Multi-body kinematics and dynamics with Lie groups 9781785482311, 1785482319

Table of contents :
1. The Displacement Group as a Lie Group 2. Dual Numbers and "Dual Vectors" in Kinematics 3. The "Transference Principle" 4. Kinematics of a Rigid Body and Rigid Body Systems 5. Kinematics of Open Chains, Singularities 6. Closed Kinematic Chains: Mechanisms Theory 7. Dynamics 8. Dynamics of Rigid Body Systems

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Multi-Body Kinematics and Dynamics with Lie Groups

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Series Editor Noël Challamel

Multi-Body Kinematics and Dynamics with Lie Groups

Dominique P. Chevallier Jean Lerbet

First published 2018 in Great Britain and the United States by ISTE Press Ltd and Elsevier Ltd

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Press Ltd 27-37 St George’s Road London SW19 4EU UK

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Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. For information on all our publications visit our website at http://store.elsevier.com/ © ISTE Press Ltd 2018 The rights of Dominique P. Chevallier and Jean Lerbet to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library Library of Congress Cataloging in Publication Data A catalog record for this book is available from the Library of Congress ISBN 978-1-78548-231-1 Printed and bound in the UK and US

Contents

List of Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

Chapter 1. The Displacement Group as a Lie Group

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1.1. General points . . . . . . . . . . . . . . . . . . . . . . . . 1.2. The groups O(E) and SO(E) as Lie groups . . . . . . . . 1.2.1. Preliminary remarks . . . . . . . . . . . . . . . . . . . 1.2.2. Elementary calculus in O(E) and SO(E) seen as manifolds . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3. Exponential mapping of SO(E) . . . . . . . . . . . . 1.3. The group U of normalized quaternions . . . . . . . . . . 1.3.1. Quaternionic representation of SO(E) . . . . . . . . 1.3.2. Complement . . . . . . . . . . . . . . . . . . . . . . . 1.3.3. Angular velocity in quaternionic representation . . . 1.4. Cayley transforms . . . . . . . . . . . . . . . . . . . . . . 1.4.1. Cayley transform defined on La (E) . . . . . . . . . . 1.4.2. Cayley transform defined on E . . . . . . . . . . . . . 1.4.3. Relation between Cayley transform and quaternions . 1.4.4. Angular velocity of a motion described with a Cayley representation . . . . . . . . . . . . . . . . . . . . 1.5. The displacement group as a Lie group . . . . . . . . . . 1.5.1. The displacement group as a matrix group . . . . . . 1.5.2. The displacement group as a group of affine maps . . 1.5.3. Classification of the Euclidean displacements . . . . 1.5.4. The Lie algebra of D as a Lie algebra of vector fields on E . . . . . . . . . . . . . . . . . . . . . . . . 1.5.5. The Klein form on D . . . . . . . . . . . . . . . . . . 1.5.6. Operator  . . . . . . . . . . . . . . . . . . . . . . . .

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1.5.7. One-parameter subgroups of D and exponential mapping 1.6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7. Appendix 1: The algebra of quaternions . . . . . . . . . . . 1.7.1. First definition of quaternions . . . . . . . . . . . . . . . 1.7.2. Center of H . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.3. Conjugation in H . . . . . . . . . . . . . . . . . . . . . . 1.7.4. Euclidean structure of H . . . . . . . . . . . . . . . . . . 1.7.5. Second definition of quaternions . . . . . . . . . . . . . . 1.8. Appendix 2: Lie subalgebras and ideals of D . . . . . . . . . 1.8.1. Lie subgroups of D . . . . . . . . . . . . . . . . . . . . . 1.8.2. Trivial Lie subgroups . . . . . . . . . . . . . . . . . . . . 1.8.3. One-parameter subgroups . . . . . . . . . . . . . . . . . . 1.8.4. Two-parameter subgroups . . . . . . . . . . . . . . . . . 1.8.5. Three-parameter subgroups . . . . . . . . . . . . . . . . . 1.8.6. Four-parameter subgroups . . . . . . . . . . . . . . . . . 1.8.7. Five-parameter subgroups . . . . . . . . . . . . . . . . .

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66 71 73 73 74 74 74 76 77 79 81 81 82 82 82 83

Chapter 2. Dual Numbers and “Dual Vectors” in Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

2.1. The Euclidean module D over the dual number ring . . . . . . . . . . . 2.1.1. The ring Δ and the module structure on D . . . . . . . . . . . . . . 2.1.2. Linear independence over Δ . . . . . . . . . . . . . . . . . . . . . . 2.1.3. Δ-linear maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4. Dual inner and mixed products . . . . . . . . . . . . . . . . . . . . . 2.2. Dualization of a real vector space . . . . . . . . . . . . . . . . . . . . . 2.2.1. General extension of a real vector space into a Δ-module . . . . . 2.2.2. Dualization of the Euclidean vector space in dimension 3 . . . . . . 2.2.3. The groups O(D) and SO(D) . . . . . . . . . . . . . . . . . . . . . 2.2.4. Generalized Olinde Rodrigues formula . . . . . . . . . . . . . . . . 2.3. Dual quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1. Geometrical definition . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2. Norm and invertibility in H . . . . . . . . . . . . . . . . . . . . . . 2.3.3. Dual quaternions and representation of D in D . . . . . . . . . . . . 2.4. Differential calculus in Δ-modules . . . . . . . . . . . . . . . . . . . . 2.4.1. Δ-differentiable maps . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2. Extensions of ordinary differentiable maps into Δ-differentiable maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86 86 88 89 90 97 97 99 101 106 110 110 111 112 113 113 114

Chapter 3. The “Transference Principle” . . . . . . . . . . . . . . . . . . 119 3.1. On the meaning of a general algebraic transference principle  . . . . 3.2. Isomorphy between the adjoint group D∗ and SO(E) 3.3. Regular maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Extensions of the regular maps from U to SO(E) . . . . . .

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Contents

vii

Chapter 4. Kinematics of a Rigid Body and Rigid Body Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Kinematics of a rigid body . . . . . . . . . . . . . . . . 4.3. The position space of a rigid body . . . . . . . . . . . . 4.4. Relations to the models of bodies . . . . . . . . . . . . 4.4.1. Example 1 . . . . . . . . . . . . . . . . . . . . . . . 4.4.2. Example 2 . . . . . . . . . . . . . . . . . . . . . . . 4.4.3. Fundamental theorem of kinematics of a rigid body 4.5. Changs of frame in kinematics . . . . . . . . . . . . . . 4.6. Graphs and systems subjected to constraints . . . . . . 4.6.1. A few elements of graph theory . . . . . . . . . . . 4.6.2. The position space of a rigid body system . . . . . 4.6.3. The various kinds of links between pairs of bodies 4.6.4. Kinematics of a linked pair of bodies . . . . . . . . 4.7. Kinematics of chains . . . . . . . . . . . . . . . . . . .

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129 131 131 134 134 135 135 137 139 139 141 143 146 148

Chapter 5. Kinematics of Open Chains, Singularities . . . . . . . . . . 153 5.1. The mathematical picture of an open chain . . . . . . . . . . 5.1.1. Articular coordinates . . . . . . . . . . . . . . . . . . . . 5.1.2. The shape function in articular coordinates . . . . . . . . 5.1.3. Further developments about the shape function . . . . . 5.2. Singularities of a kinematic chain . . . . . . . . . . . . . . . 5.2.1. General setting . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2. A method to calculate the rank of a subset of D . . . . . 5.3. Examples: Singularities of open kinematic chains with parallel axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1. Singularities of an open chain HHHH with parallel axes 5.3.2. Singularities of an open chain HPHP with parallel axes . 5.3.3. Singularities of an open chain HPPH with parallel axes . 5.3.4. Singularities of an open chain HHPP with parallel axes . 5.3.5. Singularities of open chains HHH with parallel axes . . 5.3.6. Singularities of a chain HPH with parallel axes . . . . . 5.4. Calculations of the successive derivatives of f . . . . . . . . 5.5. Transversality and singularities of a product of exponential mappings . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1. Structure of manifold of Er . . . . . . . . . . . . . . . . 5.5.2. Tangent space of Er . . . . . . . . . . . . . . . . . . . . . 5.5.3. Transversality . . . . . . . . . . . . . . . . . . . . . . . . 5.5.4. The tangent space of Σr (f ) at a weak singular point . . 5.5.5. The global features and the imperfections in the joints: case n = dim D . . . . . . . . . . . . . . . . . . .

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Chapter 6. Closed Kinematic Chains: Mechanisms Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 6.1. Geometric framework and regular case . . . . . . 6.2. Exhaustive classification of the local singularities of mechanisms . . . . . . . . . . . . . . . . . . . . . . 6.2.1. Weak singular configuration . . . . . . . . . . 6.2.2. Strong singular configuration . . . . . . . . . . 6.3. Singular mechanisms with degree of mobility one 6.3.1. The principle of the method . . . . . . . . . . 6.3.2. The Benett mechanism . . . . . . . . . . . . . 6.3.3. The Bricard mechanism . . . . . . . . . . . . . 6.4. Concrete examples and calculations . . . . . . . . 6.4.1. The Bricard mechanism . . . . . . . . . . . . . 6.4.2. The four-bar planar mechanism . . . . . . . .

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Chapter 7. Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 7.1. Changes of frame in dynamics, objective magnitudes 7.2. The inertial mass of a rigid body . . . . . . . . . . . . 7.2.1. Center of inertia . . . . . . . . . . . . . . . . . . . 7.2.2. Bilinear form associated with the kinetic energy . 7.2.3. Matrix representation of Hs . . . . . . . . . . . . 7.2.4. Kinetic and dynamic momentum . . . . . . . . . . 7.3. The fundamental law of dynamics . . . . . . . . . . . 7.3.1. Mathematical pictures of forces . . . . . . . . . . 7.3.2. Statement of the fundamental law in Galilean frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3. Energy integral . . . . . . . . . . . . . . . . . . . . 7.3.4. Law of dynamics with respect to a non-Galilean frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 8. Dynamics of Rigid Body Systems . . . . . . . . . . . . . . . 269 8.1. Systems subjected to constraints . . . . . . . . . . . 8.1.1. The position space of a rigid body system . . . 8.1.2. The various kinds of links between pairs of bodies . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.3. Forces exerted in links . . . . . . . . . . . . . . 8.2. The principles of dynamics for multibody systems . 8.2.1. Newton-Euler form of the principles . . . . . . 8.2.2. Virtual power form of the principle . . . . . . . 8.2.3. Virtual velocities in a system with a linkage . . 8.3. Tree-structured systems . . . . . . . . . . . . . . . . 8.3.1. Kinematics of a tree-structured system . . . . .

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Contents

8.3.2. Remarks . . . . . . . . . . . . . . . . . . . . . . . 8.3.3. Virtual power of deformations in a system with a linkage . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.4. Application of the principle of virtual works to a tree structured system . . . . . . . . . . . . . . . . . . 8.4. Complement: Lagrange’s form of the virtual power of the inertial forces . . . . . . . . . . . . . . . . . . . . . 8.5. Appendix: The subspaces n(s) and m(s) associated with the Lie subalgebras of D . . . . . . . . . . . . . . . . 8.5.1. Dimension 1 Lie subalgebras and Lie subgroups . 8.5.2. Dimension 2 Lie subalgebras and Lie subgroups . 8.5.3. Dimension 3 Lie subalgebras and Lie subgroups . 8.5.4. Dimension 4 Lie subalgebras and Lie subgroups . Bibliography Index

ix

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List of Notations

N: Z: R: H: Δ: : Df : fT : ◦: E: Tr (u): E: ∧:

set of integers ring of rational integers real number field real quaternion field (Appendix 1 to Chapter 1) dual number ring (see section 2.1.1) dual number such that 2 = 0 (see sections 1.5.6 and 2.1.1) differential of the map f (see section 1.1) tangent map of the map f (see section 1.1) symbol of the composition of maps Euclidean affine space (generally of dimension 3) translation in E by the vector u ∈ E (see section 1.5.3) vector space (very often the Euclidean of dimension 3 over R) vector product (or “cross product”) in the oriented dimension 3 Euclidean vector space ˜: a mapping x → a ∧ x in the oriented dimension 3 Euclidean vector space [· | ·]: Lie bracket in a Lie algebra (·; ·; ·): mixed product in the dimension 3 Euclidean vector space {· | ·}: dual mixed product in the Δ-module D (see section 2.1.4) {·; ·; ·}: dual mixed product in the Δ-module D (see section 2.1.4) L(E): algebra of the linear operators in the vector space E (see section 1.2.1) Gl(E): group of the regular linear operators the vector space E (see section 1.2.1) O(E): Orthogonal group of the Euclidean vector space E (see section 1.2) SO(E): Special othogonal group of the Euclidean vector space E (see section 1.2) U: group of the normalized quaternions Ga(E): group of affine transformations of E

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D, D(E): Displacement group of E D, D(E): Lie algebra of D (of the skewsymmetric vectorfields on E see section 1.5.4) [[· | ·]]: Klein form of D (see section 1.5.5) Z: set of vector fields ∈ D − T vanishing on their axe (see section 1.5.5) Za : set of vector fields ∈ D − T vanishing at a ∈ E (see section 1.5.5) T: ideal of D of the constant vector fields on E (see section 1.5.4). ΔX : axis of X ∈ D (see section 1.5.4) fX or pX : pitch of X ∈ D (see section 1.5.4) G: Lie group (defined according to the context) T G: tangent space of G Tg G: tangent vector space of G at g g: Lie algebra of the Lie group G Ad : adjoint representation of a Lie group (defined according to the context, see sections 1.2.2, 1.5.1,1.5.4) A∗ : equivalent to Ad A in the Euclidean displacement group (see section 1.5.2) ad : adjoint representation of a Lie algebra (defined according to the context, see section 1.2.2) ϑ  , ϑr : Left and right Maurer-Cartan forms on a Lie group (generally SO(E) or D see sections 1.2.2 or 1.5.1)

Introduction

The first significant occurrence of Lie groups in classical mechanics is due to V. Arnold in the paper [ARN 66] (1966) who studies Eulers’s equations for the dynamics of a rigid body or of a perfect fluid and points out that, up to the choice of the group, their structure is similar. Today, articles of mechanics and physics referring to Lie groups, especially in Hamiltonian dynamics or in control theory are various and very numerous. However they often focus on theoretical properties, such as integrability or reduction with the help of first integrals, of rather particular mechanical systems which are of little interest for the common engineer who encounters complicated mechanical systems and wishes to simulate their behavior with a computer. In other words, the presentation of the dynamics of a single rigid body in the light of Lie group theory is more or less classical but limited in scope so that extensions to large mechanical systems, falling in the scope of mechanical engineering, are not quite common. Certainly many articles in applied mechanics refer to Lie groups, for instance in their title, but indeed they often make no real use of the powerful mathematical techniques of algebra and differential calculus derived from the structure of Lie group as it is understood in mathematics. There are various methods to describe the configurations of rigid body systems with coordinates.Indeed they all amount to describe by one or another technique Euclidean displacements performed by the elements of the system and the significant mechanical properties are those which, after all, can be expressed in the language of this group. The goal of this book is double: first to show that the concept of Lie group can be useful to mechanical engineering, second to show that the calculations with Lie groups are powerful, very easy to handle in practical mechanical problems, on one condition: to make a small effort in order to learn some rules. The book aims at demonstrating that those required rules are not numerous, and that they make a complete system to state all the problems of general mechanics in a very compact form fully compatible with numerical or algebraic softwares. Whereas the common approach to the modeling of a complex mechanical system starts with a “forest of

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Multi-Body Kinematics and Dynamics with Lie Groups

frames”, the modelization based on Lie groups needs no necessary frame and no coordinates and is expressed in intrinsic form translatable in computer language. Concerning the first objective we may remark that, in multibody mechanics, the efficiency of Lie group calculations, in the true sense, were soon demonstrated by applications (see [MIZ 92, MIZ 88, MON 84], going with the theoretical works [LER 88, CHE 86]). The task will be to extend the Lie group language from the more or less classical applications to the mechanics of a single rigid body to systems of rigid bodies and the above mentioned rules contain all the necessary algebraic and differential calculus required to generate their kinematic and dynamic equations. In the model of the systems we consider in the book we assume that the joints may be described Lie subgroups of the Euclidean displacement group what is the most common occurrence. Of course for final numerical calculations it will be necessary to refer to some coordinate system but a big advantage of the method is that the heavy calculations may be switched to the computer; only the mathematical structure of the statement of the mechanical problem in the language of differential calculus in Lie groups will be necessary at the preliminary stages of the design of a software. The second objective is perhaps the leading motivation of this work. It was mentioned above: as soon as they concern rigid bodies, the calculations in kinematics and dynamics are calculations the Euclidean group D in dimension 3 which is a classical Lie group. From this standpoint they may be roughly distributed among three main levels according to the sharpness of the mathematical structure which is concerned at each level. – The calculations of level 1, do not refer to the particular features of the Euclidean group, they only rely on the general Lie group structure of D (that is to say only the algebraic structure of group and the structure of manifold allowing the differential calculus). – The calculations of level 2, use a particular feature of the Euclidean displacement group and they refer to the splittings of D into a “translation group” and a “rotation group” about some fixed origin in space. – The calculations of level 3 are performed in the coordinate language where all the quantities are described by matrices depending on coordinates in D (as Euler angles or Cayley-Klein parameters and so on). It is at level 2 that, in mechanics, all relation splits into a “linear part” and an “angular part”, that the velocity of a rigid object may be described by a linear and an angular velocity, that a “torsor” splits into two “Plücker’s vectors”. At this level all the calculations may be performed in dimension 3 with standard vector algebra, but the structure of all formulas are much more complicated than at level 1.

Introduction

xv

At level 1, the calculations may be performed with a well defined algebra in dimension 61. It is at this level that all the mathematical relations take the most compact form so that it is worth to perform the most possible part of calculations at this level. When it will be necessary to come back to a more familiar mathematical formalism the translation of those relations into (more complicated!) relations taking into account properties of the Euclidean group and Euclidean geometry will be easy. For instance what is at level 1 a product of two elements of a group D becomes at level 2, with a more detailed representation of those elements, a product of two 6 × 6 matrices of operators of a well defined form and, at level 3, may become a product of two 4 × 4 matrices with the well-known more detailed representation of displacements in coordinates (section 1.5.1) or another product of matrices when other representations are more convenient. As it will be explained in Chapter 1, this process of gradual translation integrates all the necessary differential calculus. The mathematical form of the relations holding at level 1 is preserved through all this process even if we should include a level for a computer language. It seems to be necessary to stress on the fact that, despite the rather abstract mathematical language used at level 1, it may be very readily translated into a programming language. A frequent criticism against the improvements of the mathematical methods to model the mechanical systems by using more sophisticated mathematics says that the reward for the necessary efforts are out of proportion to the gain of their use. If such criticism would be fully justified their would not be so many attempts to get over the difficulty to solve kinematical problems or to build the dynamic equations for many body systems. And so many articles to come back to this problem presenting attempts by means of “new methods”2. This situation points out that a need for clarification arises. It is not easy to understand the mathematical structure which is behind the dynamic equations through their the very complicated expanded form in coordinates. The clarification will likely come from using an intrinsic formalism. The effectiveness of a direct approach by Lie group and Lie algebra theory is highlighted by the complete classification of the singularities of mechanisms; certainly, this classification would be extremely difficult to point out in the coordinate language. The same remark may be done about the investigations of the mathematical structure of the dynamic equation in the line of an easy interface between mathematical modelization and computer. The organization of the book is the following: Chapter 1 introduces the Lie group structure on the various examples of groups involved in mechanics of rigid body and rigid body systems from the standpoint of algebra and differential calculus. The mathematical techniques introduced in this 1 In practice but, in fact, at level 1 the form of the calculation is independent of the dimension. 2 The complexity of this problem, and consequently the need for truly new methods, was emphasized in Y. Papegay’s thesis [PAP 92] where expanded forms of the dynamic equations which should be almost impossible to derive “by hand” are demonstrated.

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Multi-Body Kinematics and Dynamics with Lie Groups

chapter contain a complete system of rules for expanding all the calculations in kinematics and dynamics of articulated multibody systems. Those techniques are nothing but those of general theory of Lie groups applied to the Euclidean groups. Chapter 2 presents the theory of dual numbers and dual vectors in an intrinsic form, showing that it is the study of a module structure on the Lie algebra of the Euclidean displacement group. Chapter 3 completes the preceding Chapter 2 with some remarks on the so called transference principle. In Chapter 4 the book starts with mechanics proper and points out the links between kinematics and the differential calculus in Lie groups in the typical case of a rigid bodies and chains of linked rigid body. This chapter also points out the relations between the standpoint of Lie groups and the more familiar expositions of mechanics relying on the models of rigid bodies as aggregates of particles. Chapters 5 and 6 deal with kinematics of open and closed chains (i.e. mechanisms) and their singularities with examples of calculations based on the mathematical framework of this book. Chapter 7 is a detailed presentation of the dynamics of the rigid body in the frame of Lie groups and of its links with the classical presentation of this matter. The fundamental law of dynamics is presented in Galilean and non-Galilean frames directly for a realistic body (not reduced to a massive particle). Everything indicates that the full system of the dynamic equations of a rigid body – in translation and in rotation and reduced to one equation [7.34] in dimension 6 – takes a very simple mathematical form easy to handle in the framework of multibody systems. Chapter 8 deals with dynamics of rigid body systems. In particular as an example, a complete presentation of techniques to generate the dynamic equations of a treestructured system. Some exercises are proposed in order that the reader who will deal with them will become more familiar with the mathematical framework used in the book. In particular some proofs of theorems or propositions are left to the reader as exercises. An asterisk indicates an exercise or a question requiring the knowledge of rather technical tools in mathematics. Each of these chapters includes an introduction with bibliographical references. We only mention here some general points. The systematic use of the (algebraic) structure of group in mechanisms theory, widely improved in the direction of the design of robots, was introduced by J. M. Hervé [HER 78] (1978) and in [HER 82], [HER 94]. The differential calculus on Lie groups with applications to kinematics was developped by A. Karger and J. Novak [KAR 85] (1985).

Introduction

xvii

Kinematics and dynamics of multibody systems were presented in Lerbet [LER 88] (1988), Chevallier [CHE 86] (1984). Applications to dynamics of concrete complidcated mechanical systems appeared in C. Monnet and D. Chevallier [MON 84] (1984) or J.P. Mizzi [MIZ 92, MIZ 88] (1988). More recently Andreas Muller developped applications to mechanisms, with numerical algorithms (see for instance [MÜL 03, MÜL 14a, MÜL 14b]), Frederic Boyer and A. Shaukat (see [BOY 11] and [BOY 12]) developed applications to robotics fixed and mobile multibody systems including elements of Lie group theory in numerical methods.

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1 The Displacement Group as a Lie Group

In this chapter we point out the general properties of Lie groups on examples (the rotation group SO(E), the group U of normalized quaternions, the Euclidean displacement group in three dimensions described first as a set of 4 × 4 matrices and after as a group of transformations of the affine Euclidean space. We have avoided relying on the general theory of Lie groups; of course a reader who already knows this general theory will find no new information in those examples. In essence, the calculations performed in kinematics and dynamics of rigid body systems are calculations on Lie groups, nevertheless this feature is quite hidden when they are performed in the common expanded form with matrices, coordinates and so on. In the present chapter the challenge will be to show the way to perform those calculations in the language of Lie groups. The similar conclusions we shall draw from the various examples will be that the structure of this language is simple, that it relies on a few standard formulae and that it provides us with ability to operate easily on compact expressions. Regarding differential geometry and Lie groups many treatises are available. The reader may refer to the books [AND 02] by A. Baker, [AUS 77] by L. Auslander and R. MacKenzie, [CHE 06] by D. Chevallier. 1.1. General points A Lie group is a set endowed with two mathematical structures. First a structure of group in the meaning of algebra, second a structure of differentiable manifold allowing differential calculus and such that the operations in the group, namely product and inversion, are differentiable mappings. From the stand point of mathematics a differentiable manifold is a rather complicated object. However the manifolds we shall meet in the following are submanifolds of vector spaces or affine spaces, like

2

Multi-Body Kinematics and Dynamics with Lie Groups

“surfaces”; they are simple mathematical objects because we can rely on elementary results of differential geometry, the differential calculus is clear and needs no sophistication. First of all let us specify those results used in the following. The most simple example of differentiable manifold is an open subset U of a real vector space X (may be U = X itself!) and then a tangent vector v of U according to differential geometry must be considered as a pair (x, ξ) where x ∈ U is the origin of v and ξ ∈ X is the “vector” itself. The tangent space of U will be T U = U × X and the tangent vector space at x ∈ U will be Tx U = {x} × X, a vector space isomorphic with X. When two vector spaces X and Y are considered as manifolds, if U is an open subset of X and f : U → Y is a differentiable map, rather than the differential Df itself, the relevant concept in differential geometry is the tangent map f T : T U → T Y such that f T (v) = (f (x), Df (x)(ξ)). Note that f T transforms a tangent vector of U with origin x into a tangent vector of Y with origin f (x) and, in some sense, it contains both f and its differential. Another notation for tangent vectors, used by physicists and mechanicians, is v = (x, δx), then the differential is denoted by δf (v) = Df (x)(δx) and f T (v) = (f (x), δf (x)(δx)). An open subset U of affine space M over the vector space X 1, may also consider as a differentiable manifold. Then a tangent vector of U may be considered as a pair (m, ξ) with m ∈ M and ξ ∈ X and the tangent vector space at m is Tm U = {m} × X (isomorphic with X). When N is another affine space over a vector space Y and f : U → N is a differentiable map its tangent map f T : T U → T N is defined by:   f T (v) = f (m), Df (m)(ξ) for v = (m, ξ) ∈ T U i) If U is an open subset of X and f : U → Y is a differentiable function (say C k with k ≥ 2) which is a submersion at every point x ∈ U (that is to say, for all x the linear map Df (x) from X to Y is onto), then: for fixed a ∈ Y the subset M = f −1 (a) = {x ∈ U | f (x) = a} is a submanifold of U .

[1.1]

Then, a tangent vector of M with origin x is a tangent vector of X with origin x such that: v = (x, ξ) and f T (v) = 0y (the null vector of Ty Y, y = f (x)).

[1.2]

1 An affine space over X is a set M such that to any pair (m, p) of points of M is associated → ∈ X, with some properties similar to those which are given in the example a vector − mp section 1.5.2 for the space of Euclidean geometry.

The Displacement Group as a Lie Group

3

An equivalent definition says that v is a tangent vector of X which is tangent to a curve by x lying on M, that is to say that there exists a differentiable curve t → γ(t) such that γ(to ) = x and γ(t) ∈ M for all t,   dγ v = γ(to ), (t0 ) = (x, ξ). dt We shall refer to the first form of the definition of a tangent vector in sections 1.2 and 1.3, to the second one in section 1.5 to investigate the structure of the Euclidean displacement group. ii) If V is a finite dimension vector space, F : U → V is differentiable (as a mapping defined on U ), then the map induced by F on the submanifold M is differentiable (as a mapping defined on M, that is according to the general theory of differential calculus on manifolds). iii) More generally, to a differentiable mapping F : M → N of a submanifold of X into a submanifold of Y is associated a tangent mapping T F : T M → T N transforming a tangent vector v ∈ Tx M into a tangent vector T f (v) ∈ TF (x) N. T If M ⊂ U and F is induced on M by a mapping defined on U then F (x, ξ) = F (x), DF (x)(ξ) . Remark that, here we have considered that the vector spaces and the affine spaces are differentiable manifolds, that their tangent spaces are defined as products and that the tangent maps to differentiable mappings are defined by the given formulae. This is adequate for the purpose of this book. However, it would be a matter of pure mathematics to start from the specific definitions of differential geometry and to prove that these definitions agree with them. 1.2. The groups O(E) and SO(E) as Lie groups In this section we introduce the properties and the notation of Lie group theory on the example of the groups of orthogonal transformations of an Euclidean vector space.

1.2.1. Preliminary remarks Let E be a finite dimension Euclidean vector space over R, L(E) be the vector space over R of linear operators in E. Endowed with the operations of the vector space plus the product of linear operators (denoted by u.v or u ◦ v), L(E) is an associative algebra. If we define the Lie bracket by [u, v] = u.v − v.u for u and v in L(E)

4

Multi-Body Kinematics and Dynamics with Lie Groups

L(E) becomes a Lie algebra over R, that is to say (u, v) → [u, v] is a bilinear map verifying for all u, v, w in L(E) – [u, v] = −[v, u] (skew-symmetry of the bracket), – [u, [v, w]] + [v, [w, u]] + [w, [u, v]] = 0 (Jacobi identity) . We note La (E) the vector subspace of L(E) the elements of which are the skewsymmetric operators, then La (E) is a Lie subalgebra of L(E) 2. The reason why La (E) is a Lie subalgebra is that, when u∗ = −u, v ∗ = −v: [u, v]∗ = (u.v)∗ − (v.u)∗ = v ∗ .u∗ − u∗ .v ∗ = v.u − u.v = −[u, v] Let Gl(E) be the group of invertible linear operators of E, 1 its unity (identity operator). Natural subgroups of Gl(E) are associated with the Euclidean structure of E; they are: – The orthogonal group O(E) = {g ∈ Gl(E) | g ∗ .g = 1}, – The special orthogonal group SO(E) = {g ∈ Gl(E) | g ∗ .g = 1 and det g = 1}. Since the dimension of E is finite, property g ∗ .g = 1 implies g ∗ = g −1 , hence, it is equivalent to g.g ∗ = 1. Moreover, if g ∈ O(E) (resp. ∈ SO(E)) and u ∈ La (E), then g.u.g −1 ∈ La (E) because 

g.u.g −1

∗

 ∗ = g.u.g ∗ = g.u∗ .g ∗ = −g.u.g ∗ = −g.u.g −1

  (resp. and det g.u.g −1 = det(u)). Therefore we have defined a linear representation Ad of O(E) (resp. SO(E)) that is to say: – for fixed g the map u → g.u.g −1 = Ad g.u is linear from La (E) to La (E) – for g1 and g2 ∈ O(E): Ad (g1 .g2 ) = Ad g1 ◦ Ad g2 – Ad 1 = identity of La (E)   Moreover it is readily proved that Ad g.[u, v] = Ad g.u, Ad g.v , so that Ad is not only a linear representation of O(E) (resp. SO(E)) in La (E) but also a representation in the Lie algebra La (E). 2 Since E is Euclidean, the adjoint u∗ of any linear operator u ∈ L(E) is defined by u(x) · y = x · u∗ (y)

The Displacement Group as a Lie Group

5

The case where E is the three-dimension oriented Euclidean vector space, endowed with the inner product and the vector product ∧ (or cross product ×), is very important in practice. Then we have (for x, y, z ∈ E): x ∧ y = −y ∧ x,       x ∧ y ∧ z + y ∧ z ∧ x + z ∧ x ∧ y = 0 (Jacobi identity)   x ∧ y ∧ z = (x · z)y − (x · y)z (Gibbs formula)

[1.4]

g(x ∧ y) = g(x) ∧ g(y) for g ∈ SO(E)

[1.5]

[1.3]

The first two formulas mean that E is a Lie algebra over R. Jacobi identity may be ∼ derived from Gibbs formula. For a ∈ E let us define the linear operator a by ∼

a (x) = a ∧ x for x ∈ E ∼

It is readily proved that the map a → a is a Lie algebra isomorphism from E (endowed with ∧) onto La (E) (endowed with [·, ·]). In particular, Jacobi identity implies that ∼ ∼   a, b = a ∧ b)∼

[1.6]

E XERCISE 1.1.– 1) We assume that E is a real vector space endowed with an inner product x·y (non-degenerate bilinear form); E might be an Euclidean vector space but this is not necessary here. Let u : E → E be a map such that u(x) · y + x · u(y) = 0. Prove that u is a skew-symmetric linear operator. 2) If E is the oriented three-dimension Euclidean space, proved that u is of the ∼ form a. 3) Prove that, for g ∈ O(E), [1.5] spreads into the more general property: g(x ∧ y) = det(g). g(x) ∧ g(y) for x and y ∈ E.

[1.7]

H INT.– 1) Prove that the inner product of u(λx + μy) − λu(x) − μ(y) with all z ∈ E vanishes. 2) Consider the matrix of u in a right-hand orthonormal basis of E and deduce the coordinates of a in that basis. E XERCISE 1.2.– Let E be the three-dimension Euclidean vector space. Prove that the Lie subalgebras of La (E) are of three kinds: {1}, La (E) and the Lie algebras of the ∼ form Ru with u ∈ E (u = 0). ∼

H INT.– Use relation x → x and solve the problem for (E, ∧).

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Multi-Body Kinematics and Dynamics with Lie Groups

E XERCISE 1.3.– The following results will be useful in exercise 1.25, section 1.5.4, in order to find the derivations of the Lie algebra D of the Lie group D. Let (E, ∧) be the Lie algebra defined by the three dimension oriented Euclidean vector space endowed with the vector product. Recall that a derivation of this Lie algebra is a linear mapping such that A(x ∧ y) = A(x) ∧ y + x ∧ A(y). 1) Prove that the derivations of this Lie algebra are the mappings of the form a : x → a ∧ x.

∼

2) Prove that if B ∈ L(E) and L ∈ L(E) are linear operators in E then ∀x, y ∈ E : B(x) ∧ y = B(y) ∧ x ⇐⇒ B = 0 ∀x, y ∈ E : L(x ∧ y) = x ∧ L(y) ⇐⇒ L = λ1 with λ ∈ R. E XERCISE 1.4.– ∗ This exercise requires some knowledge of elementary topology in metric spaces. Remind that if E is a finite-dimension normed real vector space, the vector space L(E) of linear operators is also normed with the operator norm (taking various expressions): |f (x)| x=0 |x|

f = sup |f (x)| ≡ sup |x|=1

where | · | is the norm in E, so that L(E) becomes a metric (and topological) space with the distance d(f1 , f2 ) = f2 − f1 We now assume that E is finite-dimension Euclidean vector space, | · | is the Euclidean norm and that G is the group SO(E). 1) Check that g ∈ SO(E) ⇒ g = 1 for g ∈ G. 2) Check that d(g1 , g2 ) = g2−1 .g1 − 1 defines a distance on G and that this distance is invariant by the left and right translations in the group G (i.e. d(hg1 , hg2 ) = d(g1 h, g2 h) = d(g1 , g2 ) for all g1 , g2 , h ∈ G). 3) Check that the operations in the group G (i.e. product and inversion) are continuous maps. 4) Prove that G is a compact subset of L(E). H INT.– In a finite dimension vector space over R the compact subsets are the closed ans bounded subsets.

The Displacement Group as a Lie Group

7

1.2.2. Elementary calculus in O(E) and SO(E) seen as manifolds We only consider the case of G = SO(E) (the case of O(E) is similar suppressing the condition on the determinant and replacing Gl+ (E) by Gl(E) in the following). The groups Gl(E) or Gl+ (E) = {g ∈ Gl(E) | det g > 0} are open subsets of L(E), what is the simplest kind of manifold embeded in L(E). From the standpoint of differential geometry, the subgroup G = SO(E) appears as a submanifold embeded in the vector space L(E), or in its open submanifold Gl+ (E) and the following properties mean that it is a Lie group (see exercise 1.5): If G is considered as a manifold, the mappings associated with the algebraic operations of G, namely π : (g1 , g2 ) → g1 .g2 from G × G → G and ι : g → g −1 from G to G, are differentiable (the tangent mappings of π and ι will be expressed by [1.16] and [1.17] in exercise 1.6). When the domain of a mapping is any subset of a vector space it is not correct to speak of differentiability, the meaning of differentiability becomes clear when, for instance, the subset is a submanifold. Here π and ι are induced on a submanifold of a vector space L(E) by mappings L(E) × L(E) → L(E) and Gl(E) → Gl(E) which are known to be differentiable by standard results on linear operators or matrices and the property is automatically true as it was remarked above. T HEOREM 1.1.– Let G be either the group O(E) or SO(E) and e(= 1) be its unity. Then: i) The tangent space Te G of the submanifold G of L(E) at e is isomorphic with the vector space La (E). It may be endowed with a natural structure of Lie algebra isomorphic to the Lie algebra La (E). ii) For all g ∈ G the tangent vector space Tg G is isomorphic with Te G and with La (E) by two ways defined according to [1.10]. C OROLLARY 1.1.– If E is the oriented three-dimension Euclidean space then Te (G) is isomorphic with the Lie algebra E defined as the vector space E endowed with the vector product ∧ (or “cross product”). 2 Let Ls (E) be the subspace of the symmetric operators of L(E). Define the map f : Gl+ (E) → Ls (E) such that f (u) = u∗ .u

[1.8]

so that g ∈ G ⇐⇒ g ∈ Gl+ (E) and f (g) = 1.

[1.9]

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Multi-Body Kinematics and Dynamics with Lie Groups

We assume to be deduced precisely from [1.9] that G is a submanifold (see exercise 1.5). Then a tangent vector of L(E) with origin g is a pair (g, δg) with δg ∈ L(E) and if g ∈ G: (g, δg) ∈ Tg G ⇐⇒ f T (g, δg) = 0g (null vector with origin g) ⇐⇒ g ∈ G and Df (g)(δg) = 0 ⇐⇒ g ∈ G and g ∗ .δg + (δg)∗ .g = 0. But, since g ∗ = g −1 , when g ∈ G, it is readily proved that g ∗ .δg + (g ∗ .δg)∗ = g ∗ .δg + (δg)∗ .g = 0 ⇐⇒ δg.g ∗ + g.(δg)∗ = δg.g ∗ + (δg.g ∗ )∗ = 0 so that, finally:

(g, δg) ∈ Tg G

⎧ ⎨ ⇐⇒ g ∈ G and g −1 .δg ∈ La (E), ⎩

⇐⇒ g ∈ G and δg.g −1 ∈ La (E).

[1.10]

If we take g = e then, the conditions to the right are equivalent and (e, u) ∈ Te G ⇐⇒ u ∈ La (E). On the so defined space Te G we can define the Lie bracket by 

 (e, u), (e, v) = (e, [u, v]),

[1.11]

leading to a Lie algebra structure on Te G. If we identify {e} × La (E) with La (E) (i.e. (e, u) with u) this structure is isomorphic to that pointed up on La (E). The isomorphisms verifying ii) are exactly: ϑr : (g, δg) → g −1 .(g, δg) ≡ (e, g −1 .δg), ϑ : (g, δg) → (g, δg).g −1 ≡ (e, δg.g −1 ). and we may consider that these maps are: ϑ : (g, δg) → g −1 .δg, ϑr : (g, δg) → δg.g

−1

,

[1.12] [1.13]

The Displacement Group as a Lie Group

9

The corollary is a consequence of the isomorphy of the Lie algebras (La (E), [·, ·]) ∼ and (E, ∧) in dimension 3, every (e, u) ∈ Te G, u is of the form ω and it is sufficient to refer to [1.6] and [1.11]. D EFINITION 1.1 (Lie algebra of G = O(E) or SO(E)).– The tangent space Te G endowed with the Lie bracket [1.11] is the Lie algebra of the group G and is denoted by g3. The maps ϑr and ϑ from T G to g defined in [1.12] and [1.13] are the left and right Maurer-Cartan forms of G. For the classical groups other notations are commonly used: The Lie algebra of the group Gl(E) is denoted by gl(E) (and it is isomorphic with the vector space L(E) endowed with the commutator as Lie bracket, see section 1.2.1). The Lie algebras of O(E) and SO(E) are identical and also denoted by so(E) (or o(E)) and, as we have seen, they are isomorphic with the vector space La (E) endowed with the commutator. From the standpoint of differential geometry, Maurer-Cartan forms are g-valued differential forms on the manifold G. They are essential to the mathematical expression of kinematics and dynamics in the framework of Lie groups where a lot of classical quantities may be interpreted as values of those forms. Indeed, Maurer-Cartan forms provide us with the means to avoid difficulties with the differential calculus on the tangent space T G (remind that dynamics uses second order derivatives....). An equivalent way of studying the tangent space of G would be to define the tangent vectors of G with origin g as the tangent vectors of L(E) which are tangent to the curves by g lying on G (see section 1.1). This method meets the standard situation classical kinematics: with more familiar notations, amotion of rotation of a body will be described by a derivable map t → R(t) ∈ SO(E) = G and it is well-known that, at every time, there exists vectors ω (angular velocity seen “with respect to a space fixed frame”) and Ω (angular velocity seen “with respect to a body fixed frame”) such that ∼

∼ R˙ = ω ◦R and R˙ = R ◦ Ω .

[1.14]

The proof of existence of ω and Ω in kinematics is nothing  more than  the proof ˙ of [1.10]. In fact, within the general notation, at every time R(t), R(t) ∈ TR(t) G, and, with [1.12] and [1.13] the skew-symmetric operator associated to the angular 3 In general the Lie algebra of a Lie group G denoted by the same letter in gothic script g. It is the tangent vector space at identity e endowed with a canonical Lie bracket.

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Multi-Body Kinematics and Dynamics with Lie Groups

velocities may be interpreted as values of the Maurer-Cartan forms on this tangent vector:   ∼ ˙ ω(t) = ϑr R(t) ,

∼   ˙ Ω(t) = ϑ R(t) .

The first relation means that ω is the angular velocity with respect to a “space fixed frame” (Eulerian standpoint) and the second means that Ω is the angular velocity seen with respect to a “body fixed frame” (Lagrangian standpoint). This interpretation of angular velocities ω and Ω in the framework of Lie groups was a key idea in article [ARN 66] by V. Arnold about Euler’s equations of rigid body dynamics. Let us come back to general notation and introduce another form of the “adjoint” representation introduced  in section 1.2.1. If (e, u) ∈ Te G and g ∈ G then g.(e, u).g −1 = e, g.u.g −1 ∈ Te G. D EFINITION 1.2 (Adjoint representation of G = O(E) or SO(E)).– The adjoint  representation of G is the map Ad : G → L(g) such that Ad g.(e, u) = e, g.u.g −1 . The characteristic properties of a linear representation are readily verified: – for fixed g, Ad g is a linear operator in Te G, – Ad e = identity of Te G – for g1 and g2 ∈ G, Ad g1 .g2 = Ad g1 ◦ Ad g2 . Moreover; it is easily checked that, for (e, u) and (e, v) ∈ Te G:   Ad g.[(e, u), (e, v)] = Ad g.(e, u), Ad g.(e, v) .

[1.15]

Property [1.15] means that adjoint representation is not only a linear representation, but also a representation in the Lie algebra g of the group. The following relation between Maurer-Cartan forms is evident ϑr (g, δg) = Ad g.ϑ (g, δg) for all tangent vector (g, δg). In order to explain the notation of the following theorem recall that, in general, the adjoint representation ad of a Lie algebra g is the map g → L(g) defined by: ad u.x = [u, x] and that an equivalent form of Jacobi identity means that ad u is a derivation of g:     ad u.[x, y] = ad u.x, y + x, ad u.y .

The Displacement Group as a Lie Group

11

P ROPOSITION 1.1 (Differential of adjoint representation of G = O(E) or SO(E)).– The differential of the map g → Ad g at g may be expressed by two ways: D(Ad g)(v) = ad ϑr (v) ◦ Ad g = Ad g ◦ ad ϑ (v) for v ∈ Tg G. or, what is equivalent, for fixed u ∈ g:     D(Ad g.u)(v) = ϑr (v), Ad g.u = Ad g. ϑ (v), u 2 If u = (e, u) and v = (g, δg), it is equivalent to prove the two equalities     D(Ad g.(e, u))(g, δg) = ϑr (g, δg), Ad g.(e, u) = Ad g. ϑ (g, δg), (e, u) By definition Ad g.(e, u) = (e, g.u.g −1 ) = (e, g.u.g ∗ ) and 

 D g.u.g (g, δg) = g.u.(δg)∗ + δg.u.g ∗ = ∗



g.u.g −1 g.(δg ∗ ) + δg.g −1 .g.u.g −1 g.u.(δg)∗ .g.g −1 + g.(g −1 .δg).u.g −1

Since g ∗ = g −1 and since operators δg.g −1 and g −1 .δg are skew symmetric by Theorem 1.1, we may read  ∗  ∗ g.(δg)∗ = δg.g ∗ = δg.g −1 = −δg.g −1  ∗  ∗ (δg)∗ .g = g ∗ .δg = g −1 .δg = −g −1 .δg Hence, with the definition [1.11] of the Lie bracket as a commutator: ⎧  ⎨ δg.g −1 , g.u.g −1  D g.u.g ∗ (g, δg) =  ⎩  −1 g. g .δg, u .g −1 

what is equivalent to the result to be proved. E XERCISE 1.5.– Give a formal proof that O(E) and SO(E) are submanifolds imbeded into the vector space L(E) and that it is a Lie group (i.e. operations are differentiable maps). H INT.– First of all remind the definition of a submanifold in differential geometry and the criterion mentioned in section 1.1. Remind that Gl(E) is an open subset of L(E) broken up into two open subsets {det(u) > 0} and {det(u) < 0}, and that SO(E) = {g ∈ O(E) | det g > 0}. Prove that the map f (u) = u∗ .u defined as in [1.8] is a submersion (i.e. at each point u ∈ Gl(E) the differential Df (u), is a linear map from L(E) onto Ls (E)).

12

Multi-Body Kinematics and Dynamics with Lie Groups

E XERCISE 1.6.– Let us define the left (resp. right) translation by h in the group G by Lh : g → h.g (resp. Rh : g → g.h). We note the values of the tangent mapping LTh (resp. RhT ) on the tangent vector v = (g, δg) by (g, δg) → h.v = (h.g, h.δg) (resp. (g, δg) → v.h = (g.h, δg.h). 1) Check that, when v ∈ Tg G: ϑr (v) = LTg−1 (v) and ϑ (v) = RgT−1 (v) and that ϑr (h.v) = Ad h.ϑr (v), ϑr (v.h) = ϑr (v), ϑ (h.v) = ϑ (v), ϑ (v.h) = Ad h−1 .ϑr (v), (in particular ϑ and ϑr are respectively invariant by the left and right actions of G on T G). 2) Prove that if v, v1 and v2 are tangent vectors of G with respective origin g, g1 and g2 , then: π T (v1 , v2 ) = (g1 .g2 , δg1 .g2 + g1 .δg2 ) = v1 .g2 + g1 .v2 ,   ιT (g, δg) = g −1 , −g −1 .δg.g −1 = −g −1 .v.g −1 .

[1.16] [1.17]

3) Prove that Ad g is the linear map induced on g by LTg ◦RgT−1 (what is the general definition of adjoint representation of a Lie group).   ˙ . E XERCISE 1.7.– Show, by direct calculation that equation [1.14] implies ω˙ = R Ω Show that the latter property is also a consequence of Proposition 1.1.   ˙ H INT.– Use composite mapping theorem with t → Ad R(t).ϑ R(t) take the derivative and translate into the notation of [1.14].

1.2.3. Exponential mapping of SO(E) In a three-dimension Euclidean vector space E, if u is a normalized vector, Ru is  ⊥ an axis in E and Ru is the orthogonal plane (oriented by u). When R ∈ SO(E), and R(u) = u, the orthogonal plane is globally invariant under the action of R. The rotation R(u, θ) of an angle θ about axis Ru is defined as the linear map R ∈ SO(E) verifying two conditions: R(u) = u and R induces a rotation with angle θ on the  ⊥ oriented plane Ru , (R = 1 may be considered as a rotation with zero angle about any axis...). An analytic expression of R(u, θ) is given by the classical Olinde Rodrigues formula: R(u, θ) = x + sin θ u ∧ x + (1 − cos θ) u ∧ (u ∧ x).

[1.18]

The Displacement Group as a Lie Group

13

2 A proof of [1.18] amounts to check that the right hand side is a linear operator  ⊥ agreeing with R(u, θ) on both subspaces of the direct sum Ru ⊕ Ru . The representation of a rotation by Olinde Rodrigues formula is not unique: 



R(u, θ) = R(u , θ ) ⇐⇒

θ and θ ∈ 2π Z, or θ and θ ∈ / 2π Z, u = u, θ = θ (mod. 2π) [1.19]  or θ and θ ∈ / 2π Z, u = −u, θ = −θ (mod. 2π)

When θ and θ ∈ 2πZ both rotations equal 1. In particular, when the direction of the axes is the same: R(u, θ) = R(u, θ ) ⇐⇒ θ = θ

(mod. 2π),

R(u, θ) = R(−u, θ ) ⇐⇒ θ = −θ

(mod. 2π).

[1.20] [1.21]

2 Equivalence [1.19] is intuitively true. A formal demonstration of [1.20] may start with the proof of [1.20] and [1.21]. For instance the equality to the left in [1.21] means that: ∀x ∈ E

(sin θ + sin θ) u ∧ x + (cos θ − cos θ) u ∧ (u ∧ x) = 0.

The vectors u ∧ x and u ∧ (u ∧ x) are orthogonal and, since u = 1, except for particular choices of x they do not vanish. Hence the equality is equivalent to sin θ = − sin θ and cos θ = cos θ and the proof of [1.21] is over. The proof of [1.20] follows the same way. Property [1.19] is a consequence of [1.20] and [1.21]. The assumption to the left in [1.19] implies R(u, θ)(u ) = u (= R(u , θ )(u )) what is meaning sin θ u ∧ u + (1 − cos θ) u ∧ (u ∧ u ) = 0. If u ∧ u = 0, since u = u = 1, either u = u or u = −u and we conclude by [1.20] or by [1.21]. If u ∧ u = 0 then so u ∧ (u ∧ u ) = 0. Because these vectors are orthogonal, sin θ = 0 and 1 − cos θ = 0 therefore θ ∈ 2πZ and, by the same way, θ ∈ 2πZ. Implication ⇒ in [1.19] is proved. Implication ⇐ is clear. The exponential mapping leads to another expression of formula [1.18]. In general, if E is a finite dimension vector space, the exponential exp : L(E) → Gl(E) is defined by: exp u = 1 +

∞

1 n u , n! n=1

u ∈ L(E),

14

Multi-Body Kinematics and Dynamics with Lie Groups

where the series to the right is convergent for any norm on the finite dimension vector space L(E). Another definition, related to differential equations and one-parameter groups,  says that  exp u is the value for t = 1 of the unique maximal solution t → F (t) ∈ L(E) to the Cauchy problem: d F (t) = u.F (t), dt

F (0) = 1,

[1.22]

(it is possible to prove that the maximal solution is defined for t ∈ R). With one or the other definition, it is also readily proved that exp u, a priori belonging to L(E), do takes its value in Gl(E) and that (exp u)−1 = exp(−u)). Coming back to the three-dimension Euclidean vector space E we have the following result: T HEOREM 1.2.– For ω ∈ E: ⎧ ∼ when ω ∈ 2πN (in particular when ω = 0), ⎪ ⎨ exp ω = 1 ∼ ⎪ ⎩ exp ω = R(u, θ) with θ = ω and u =

ω when ω = 0.

ω

2 We refer to the definition of exponential by series. When ω = 0 it is clear that ∼ exp ω= exp 0 = 1. If ω = 0 let us define u as in the theorem and let Ru = {λu | λ ∈ R} be the line generated by u and (Ru)⊥ be its orthogonal complement. ∼

∼

– If x0 ∈ Ru we have ω (x0 ) = 0 therefore (exp ω)(x0 ) = x0 . – If x1 ∈ (Ru)⊥ we have ω ∧ (ω ∧ x1 ) = − ω 2 x1 therefore for p ∈ N: ∼

(ω)2p (x1 ) = (−1)p ω 2p x1 ,

∼

(ω)2p+1 (x1 ) = (−1)p ω 2p+1 u ∧ x1 ,

so that: ∞ ∞  

(−1)p (−1)p ∼ (exp ω). x1 = 1 +

ω 2p x1 +

ω 2p+1 u ∧ x1 (2p)! (2p + 1)! p=1 p=0

= cos θ x1 + sin θ u ∧ x1 . ∼

Therefore the linear operators exp ω et R(u, θ) agree on each of the suplementary subspaces Ru and (Ru)⊥ , hence they agree on E. E XERCISE 1.8.– Check that for G = SO(E) and g = so(E):   exp Ad g.u = Intg.(exp u) for u ∈ g, g ∈ G. and translate this result into a property of rotations (see Theorem 1.2).

[1.23]

The Displacement Group as a Lie Group

15

E XERCISE 1.9.– 1) Let t → F (t) be a map from R to G, F˙ its derivative with respect to t. Prove that the following properties, meeting the general definition of the exponential map valid in all Lie group, where e (= 1) is the unity of G, are equivalent: F (t) = exp(tu) for all t ∈ R,

  t → F (t) is the maximal solution of ϑ F˙ (t) = u,   t → F (t) is the maximal solution of ϑr F˙ (t) = u,

F (0) = e, F (0) = e.

2) Remind that ad u is the linear map v → [u, v] from g to g. Prove that Ad (exp u) = exp(ad u) for u ∈ g.

[1.24]

(an equality in the linear group Gl(g) of the vector space g  La (E) where to the right, we have the exponential in L(g) and to the left the exponential of G). Translate the result into properties of rotations (see Theorem 1.2).   H INT.– For 2), consider the mapping t → Ad exp(tu) , use Proposition 1.1 to calculate its derivative and question 1). E XERCISE 1.10 (Euler’s angles).– Let (i, j, k) be an orthonormal basis of E (Euclidean of dimension 3) and ψ, ϑ, ϕ be real variables (angles) and put (if the process is not clear draw a picture!): A = Rot(k, ψ),

Io = A(i),

Jo = A(j),

B = Rot(Io , ϑ),

C = Rot(K, ϕ),

I = C(Io ),

J = C ◦ B(Jo ),

G(ψ, ϑ, ϕ) = C ◦ B ◦ A (∈ SO(E))

K = B(k),

(Better notation would be A(ψ), B(ψ, ϑ), R(ψ, ϑ, ϕ)...,J(ψ, ϑ, ϕ)!). Angles ψ, ϑ, ϕ are Euler’s angles of the basis (I, J, K) with respect to (i, j, k). 1) Check that J = K ∧ I and (I, J, K) is an orthonormal basis of E. Find the 3 × 3 matrix Γ(ψ, ϑ, ϕ), describing G(ψ, ϑ, ϕ) in the basis (i, j, k), and such that ⎡ ⎤ ⎡ ⎤ I i ⎣ J ⎦ = Γ(ψ, ϑ, ϕ) ⎣ j ⎦ K k 2) Prove that G(ψ, ϑ, ϕ) = Rot(k, ϕ) ◦ Rot(i, ϑ) ◦ Rot(k, ϕ). (Note that, now, the axes of rotation are independent of the values of the angles). 3) Assume that ψ, ϑ, ϕ are functions of time, so that a mapping t → R(t) = G(ψ(t), ϑ(t), ϕ(t)) defines a curve in SO(E) (and a rotating motion). Find ω(t) and Ω(t).

16

Multi-Body Kinematics and Dynamics with Lie Groups

4)* Define domains in R3 , so large as possible, such that on that domain the mapping G defines a chart of the manifold SO(E). (What is happening for θ = 0 or nπ with n ∈ Z). 5) Show that the result of 2) may also be deduced from the general property of Lie groups [1.23]. ˙ ϑ, ˙ ϕ˙ with coefficients belonging to E H INT.– 3) ω and Ω are linear functions of ψ, and depending on (ψ, ϑ, ϕ). 4) Look at the points where the differential DG(ψ, ϑ, ϕ) is not regular as linear map defined in R3 . E XERCISE 1.11.– Let G be the group SO(E) and g = so(E) be its Lie algebra; We define a Lie subgroup of G as a subgroup H of G which is also a submanifold of G. Then H is also a Lie group (for the induced manifold structure). 1) Prove that the Lie algebra h of H is a Lie subalgebra of g. 2) Assuming that dim(E) = 3, prove that the Lie subgroups of G = SO(E) are of three kinds: {e}, G and the rotation subgroups about lines in E (that is to say the subgroups of the rotations R(u, θ) such that g(u) = u, which also are the oneparameter subgroups t → exp(tu) with u ∈ La (E)). H INT.– For 1) consider h as a set of tangent vectors (e, u) of curves by e and lying in H and use Proposition 1.1 at g = e. For 2), see exercise 1.2, section 1.2.1. 1.3. The group U of normalized quaternions From now on, we assume that E is a three-dimension oriented Euclidean vector space endowed with scalar product (or “dot product”) denoted by · and vector product (or “cross product”) denoted by ∧. Let H be the field of quaternions and H∗ be the group of non-zero quaternions (see Appendix). H may also be seen as either a vector space or an algebra over R. As a vector space, the set of vectorial quaternion [E] = {(0, x) = [x] | x ∈ E} is isomorphic with E. Because, if q1 = [x1 ], q2 = [x2 ] we have q1 q2 − q2 q1 = 2 [x1 ∧ x2 ], endowed with the commutator, [E] becomes a Lie algebra isomorphic with (E, ∧) (the factor 2 is unimportant). We are interested in the subgroup U of H∗ of normalized quaternions: U = {q ∈ H | |q| = 1} = {q ∈ H | q.q = 1}. As a topological space, U is homeomorphic to the sphere S3 in R4 : if an orthonormal basis is chosen in E, U may be identified with the set of (q0 , q1 , q2 , q3 ) ∈ R4 such that q02 + q12 + q22 + q32 = 1.

The Displacement Group as a Lie Group

17

T HEOREM 1.3.– Let U be the group of normalized quaternions and e (= (1, 0)) be its unity. Then: i) U is a submanifold of H and a Lie group. ii) As a vector space, the tangent space Te U is isomorphic with the subspace [E] of vectorial quaternions. It may be endowed with a Lie algebra structure isomorphic with that of [E]. iii) For all q ∈ U the tangent vector space Tq U is isomorphic to [E] by two ways defined according to [1.27] below. 2 The formal proof of i) is the matter of exercise 1.12. Therefore, the tangent space of the manifold U may be considered as a subspace of U × H. For all q ∈ U the property v = (q, δq) ∈ Tq U is equivalent to any of the following equivalent properties ¯ .δq ∈ [E], q ∈ U and δq.¯ q ∈ U and q q ∈ [E], [1.25] q ∈ U and q.δq ∈ [E], q ∈ U and δq.q ∈ [E]. q ∈ U means that f (q) = 1 where f : H∗ → H is the map such that f (q) = q.q. Taking the tangent map: (q, δq) ∈ Tq U ⇐⇒ f T (q, δq) = 0 ⇐⇒ q ∈ U and q.δq + δq.q = 0. Since δq.q and q.δq are conjugate, the right hand equality means that their real part vanishes, proving the first and fourth property [1.25]. Moreover, for q ∈ U, q = q−1 , hence q.δq + δq.q = 0 ⇐⇒ δq.q + q.δq = 0 and the second and third property [1.25] are proved by the same way. When q = 1 the four properties amount to one only (1, δq) ∈ Te U ⇐⇒ δq ∈ [E] and the isomorphy of Te U and E as vector spaces is evident. A bracket defining a Lie algebra structure on Te U is then defined by 

 (1, u), (1, v) = (1, u.v − v.u) (with vectorial quaternions u, v)

[1.26]

and the isomorphy of Lie algebras mentioned in ii) follows. According to [1.25] the maps verifying iii) are ⎧ ⎨ ϑ : (q, δq) → (1, q−1 .δq)  q−1 .δq, ⎩

ϑr : (q, δq) → (1, δq.q−1 )  δq.q−1 .

[1.27]

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Multi-Body Kinematics and Dynamics with Lie Groups

D EFINITION 1.3.– The Lie algebra defined as Te U endowed with the bracket defined in [1.26] is the Lie algebra of U and will be denoted by U. The maps ϑ and ϑr are the left and right Maurer-Cartan forme of U. As in section 1.2.2 we can define the adjoint representation of the group U in its Lie algebra as a map U → L(U) such that   Ad q.(1, u) = 1, q.u.q−1 .

[1.28]

The relation between Maurer-Cartan forms, the properties of a representation (in the Lie algebra U ), the formula for the derivative of adjoint representation are easily verified and they are similar to those we found in section 1.2.2 for SO(E), in particular in Proposition 1.1: ϑr (q, δq) = Ad q.ϑ (q, δq),   Ad q1 q2 = Ad q1 ◦ Ad q2 , Ad 1 = identity,     Ad q. (1, u), (1, v) = Ad q.(1, u), Ad q.(1, v) ,   D Ad q (v) = ad ϑr (v) ◦ Ad q = Ad q ◦ ad ϑ (v) for v ∈ Tq U, or, what is equivalent:       D Ad q.u (v) = ϑr (v), Ad q.u = Ad q. ϑ (v), u for u ∈ U, v ∈ Tq U. E XERCISE 1.12.– Give a formal proof that U is a submanifold imbeded into the vector space H and that U is a Lie group (i.e. the product and inverse in U are differentiable mappings). H INT.– H∗ is an open subset of H (for the natural topology of H). The map f : H∗ → H such that f (q) = q.¯ q is a submersion at each point q ∈ H∗ . E XERCISE 1.13.– Prove that the formulae of exercise 1.6 are also valid for the group U. E XERCISE 1.14.– Prove the formulae giving the differential of Ad q.

1.3.1. Quaternionic representation of SO(E) The groups U and SO(E) are linked by a remarkable homomorphism (not an isomorphism!) allowing to express rotations by means of quaternions and products

The Displacement Group as a Lie Group

19

of rotations by means of products of quaternions. If q ∈ U, let us define ϕq : H → H by: ϕq (y) = qy¯ q

(product of quaternions).

This map possesses the following evident properties: ϕq is a R-linear isometry: ϕq (λy + μz) = λϕq (y) + μϕq (z), λ, μ ∈ R, |ϕq (y)| = |y|

[1.29] [1.30]

ϕq maps [E] onto [E] and transforms each real quaternion into itself. [1.31] 2 Property [1.29] is evident, [1.30] is verified because the norm of a product of quaternions is independent of the order of factors. The real part of a product of quaternions is independent of the order of factors, therefore Re (ϕq (y)) = Re (qy¯ q) = Re (¯ qqy) = Re (y) hence, if y ∈ [E], then Re (y) = 0 and ϕq (y) ∈ [E]. If y is a real quaternion it is in the center of H, hence ϕq (y) = qy¯ q = q¯ qy = y. P ROPOSITION 1.2.– For all q ∈ U, there is a uniquely defined operator R(q) ∈ SO(E) such that: ∀x ∈ E : [R(q)(x)] = ϕq ([x]) ≡ q[x]¯ q.

[1.32]

The map q → R(q) is an homomorphism of groups of U onto SO(E): if q and q ∈ U R(q) ◦ R(q ) = R(qq ),

[1.33]

R(q)−1 = R(q).

[1.34]

2 The existence of the linear operator R(q) follows from [1.31]. We have to prove that R(q) ∈ SO(E). Let y, y , y be the respective images by R(q) of x, x , x ∈ [E]. Then y, y , y ∈ [E] by [1.31] and: y · y  = x · x ,

y · (y ∧ y ) = x · (x ∧ x ),

a consequence of general formulae [1.116] of Appendix for the vectorial quaternions. For example: y · y = −Re ([y][y ]) = −Re (q[x][x ]¯ q) = −Re (¯ qq[x][x ]) = −Re ([x][x ]) = x · x .

20

Multi-Body Kinematics and Dynamics with Lie Groups

The linear operator R(q), preserves the dot product and the mixed product of E, therefore it belongs to SO(E). For q, q ∈ U     [R(q) ◦ R(q )(x)] = q. q .[x].q .q = (qq ).[x].(qq ) = Rqq (x) , proving [1.33]. Since q−1 = q for q ∈ U, we have [1.34] P ROPOSITION 1.3.– If q = (s, v) ∈ U then ∀x ∈ E : R(q)(x)

⎧ ⎨ = x + 2sv ∧ x + 2v ∧ (v ∧ x), ⎩

= (1 − 2v2 )x + 2sv ∧ x + 2(v · x)v. 

If, more specifically, q = then: R(q) = R(u, θ)

[1.35]

 θ θ cos , sin u with u ∈ E, u = 1 and θ ∈ R, 2 2

(rotation of angle θ about the axis Ru).

[1.36]

2 Under the assumptions of [1.35], an easy calculation, using relations s2 + v2 = 1 and v ∧ (v ∧ x) = (v · x)v − v2 x, leads to: q[x]¯ q = [s2 x + (v · x)v + 2sv ∧ x + v ∧ (v ∧ x)] = [x + 2sv ∧ x + 2v ∧ (v ∧ x)] so that the first expression [1.35] is proved. The second expression is derived with Gibbs formula. Putting the particular expression of q with θ in this formula one finds formula [1.18]. q ∈ U and R(q) = 1 ⇐⇒ q = ±1,

[1.37]

q and q ∈ U and R(q) = R(q ) ⇐⇒ q = ±q .

[1.38]

2 When R(q) = 1, ϕq fixes the elements of [E] (see. [1.32]) and also those of [R] (see. [1.31]). Therefore ϕq is the identity of H and: ∀ y ∈ H : qy = yq that is q belongs to the center of H which is [R]. Since q ∈ U we have q = ±1, hence [1.37]. Relation R(q) = R(q ) is equivalent to R(qq ) = 1 according to [1.34] and ¯ q = ±1. Because q ∈ U we conclude that q = ±q proving [1.38]. also to q All in all the following theorem is proved:

The Displacement Group as a Lie Group

21

T HEOREM 1.4 (Representation of SO(E) by normalized quaternions).– The map q → R(q) defined by [R(q)(x)] = q[x]¯ q is a surjective homomorphism of the group U onto the group SO(E). The kernel of this homomorphism is {+1, −1} and the inverse image in U of a rotation R ∈ SO(E) is a set {q, −q} of two opposite normalized quaternions.

1.3.2. Complement Theorem 1.4 gives a typical example of two Lie groups with isomorphic Lie algebras which are not isomorphic groups: “over” any rotation, there are two quaternions and, clearly, the group U is a twofold covering of SO(E) and, since U is connected and simply connected, it is a representation of the universal covering of SO(E), (according to general topology every Lie group has a universal covering which is also a Lie group. See for example [CHE 06], Chap IX and XIII). In this section we shall study more accurately the relation between q ∈ U and R ∈ SO(E) with respect to continuity and topology. Such considerations are important in some parts of physics but they will be of little use in the following and the reader may skip to the following section if he wants to do this. The spaces U and SO(E) are metric spaces; the distances on SO(E) and U are induced by the operator norm associated to the Euclidean norm | · | of E on L(E) and by the norm of H: d(R, R ) = R − R = sup |R(x) − R (x)|, |x|=1

d(q, q ) = |q − q |. L EMMA 1.1.– If R is a rotation, θ is its angle and q ∈ U is a quaternion such that R(q) = R then:

R − 1

⎧  ⎨ = 2(1 − cos θ), ⎩

= 2 |Ve (q)| = 2

 1 − Re (q)2 .

In particular, if R ∈ SO(E), the following properties are equivalent: – R is a U-turn (θ = π up to 2kπ with k ∈ Z), – R − 1 = 2, – Every q ∈ U such that R = R(q) is a pure quaternion.

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Multi-Body Kinematics and Dynamics with Lie Groups

2 If R is expressed by [1.18] and x1 is the orthogonal projection of x on the orthogonal plane to u: R(x) − x = sin θ u ∧ x1 + (cos θ − 1) x1 . The vectors u∧x1 et x1 are orthogonal and their norms are equal. With Pythagoras theorem:  

R(x) − x = |x1 | 2(1 − cos θ) ≤ |x| 2(1 − cos θ). Since this equality is verified when x is orthogonal to u, the first formula follows. The quaternions q ∈ U such that R(q) = R are (see Proposition 1.3): 

 θ θ q = ± cos , sin u , 2 2 2

and then: 2(1 − cos θ) = 4 (sin θ/2) = 4 |Ve (q)|2 = 4(1 − Re (q)2 ) and the second formula follows. Equivalence of the three properties is now evident. Therefore, putting: N = {q ∈ U | Re (q) > 0},

O = {R ∈ SO(E) | R − 1 < 2},

according to the lemme, O is the complement in SO(E) of the set of U-turns and the relation q → R(q) induces a bijection from N onto O (according to Theorem 1.4, every R ∈ O is determined by two quaternions q et −q with non vanishing real part and one and only one of them is in N ). Moreover, by definition, N and O are open sets in U and SO(E) respectively. L EMMA 1.2.– Let a ∈ U and aN = {ay | y ∈ N }. Then ∀q ∈ aN : ∀q∈N :

√

2 |q − a| < R(q) − R(a) ≤ |q − a|,

√

2 |q − 1| < R(q) − 1 ≤ 2 |q − 1|,

(in fact, the right hand inequalities hold for all q ∈ U)). 2 The first inequalities will be deduces from the second ones. Let us prove these second inequalities. If q = (s, v) ∈ N there is s2 + v2 = 1 and 0 < s = Re (q) ≤ 1, hence: |q − 1| =

  (s − 1)2 + v2 = 2(1 − s).

The Displacement Group as a Lie Group

23

However   √ √ 2 1 − s2 = 2 (1 − s)(1 + s) = 2 | q − 1 | 1 + s Taking account of Lemma 1.1:  √ 2 | q − 1 |< 2 1 − s2 = R(q) − 1 ≤ 2|q − 1|. Because for a ∈ U, R(a)−1 = R(¯ a) we have R(q) − R(a) = R(a) ◦ (R(¯ a) ◦ R(q) − 1) = R(a) ◦ (R(¯ aq) − 1) Because R(a) is an isometry we deduce R(q) − R(a) = R(¯ aq) − 1 . Likewise, relation q − a = a(¯ aq − 1) implies |q − a| = |¯ aq − 1|. Now if ¯q ∈ N it suffices to apply the second inequalities with the quaternion a ¯q q ∈ aN , a to complete the proof of Lemma 1.2. The results we have proved are summarized in the following theorem: T HEOREM 1.5.– The relation q → R(q) induces an homeomorphism from N onto O (that is to say a bijection which is continuous so its reciprocal). More generally if a ∈ N et A = R(a), this relation induces an homeomorphism from the open set aN onto AO. 2 The maps q → aq and R → A ◦ R are isometric bijections from U onto U and from SO(E) onto SO(E) respectively. Therefore they transform open  sets into open sets. In particular aN and AO are open. If b ∈ aN then bN aN is an open neighborhood of b in N where, according to Lemma 1.2: √

2|q − b| < R(q) − R(b) ≤ 2|q − b|.

Continuity at b of the map q → R(q) follows from the right-hand side inequality, continuity at R(b) of its reciprocal map follows from the left-hand side inequality. The first assertion is the particular case where b = 1. Note that N and, more generally aN , are maximal open domains of U where the map q → R(q) is injective (all greater open domain would contain a pair of opposite quaternions). Moreover, to every A ∈ SO(E) may be associated an open neighborhood of A, for instance AO, and two open subsets aN and −aN in U, symmetric one to the other, such that the “projections” aN → AO and −aN → AO are homeomorphisms (property of a twofold covering of SO(E)).

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Multi-Body Kinematics and Dynamics with Lie Groups

1.3.3. Angular velocity in quaternionic representation Let us consider a rotation depending on time t and its quaternionic representation: R(t) ≡ R(q(t)) with q(t) ∈ U We assume that all the mappings are at least twice differentiable and we omit to mention the dependence on t. Taking the derivative of relation qq = 1 (resp. qq = 1), ˙ = 0 (resp. qq ˙ + qq˙ = 0). This relation means that expressing that q ∈ U: qq˙ + qq ˙ (resp. qq) ˙ vanishes, hence, that it is in [E] and that, the real part of the quaternion qq at each time t, there is a vector ω = ω(t) (resp. Ω = Ω(t)) such that: 1 1 [ω], qq˙ = [Ω] 2 2 1 1 q˙ = [ω]q, q˙ = q [Ω] 2 2

˙ = qq

[1.39] [1.40]

Relations [1.39] and [1.40] are equivalent because qg = q−1 . Moreover ¯ ), ω˙ = 2Ve (¨ qq

˙ = 2Ve (¯ ¨ ), Ω qq

[1.41]

It remains to be proved that ω and Ω, defined according to [1.39], are respectively the common angular velocities with respect to a “space fixed frame” (Eulerian standpoint) with respect to a “body fixed frame” (Lagrangian standpoint) of kinematics (see [1.14]) namely that: ⎧ ⎨ = ω ∧ R(q)(x). d ∀x ∈ E : R(q)(x) = ⎩ dt = R(q)(Ω ∧ x) 2 Using [1.39] and the definition of R(q), for all fixed x ∈ E, we derive: d 1 1 d [R(q)(x)] = q[x]q = [ω]q[x]q − q[x]q[ω] dt dt 2 2 1 1 = [ω][R(q)(x)] − [R(q)(x)][ω] = [ω] [R(q)(x)] 2 2 what is equivalent to the first relation to establish. Taking the derivative of ω = d  d ˙ q) . Now (q¯ ˙ q) = ˙ (a relation equivalent to [1.39]): ω˙ = 2Ve (q¯ 2Ve (qq) dt dt ¨q ¯ + q˙ q˙ and, since the second quaternion to the right is real we find the first q relation [1.41]. The proof of relations regarding Ω follows the same way.

The Displacement Group as a Lie Group

25

E XERCISE 1.15.– Use formulae [1.39] and [1.41] to calculate the angular velocity of a rotation defined by Olinde Rodrigues (formula [1.18]) and its time derivative.  H INT.– The quaternion defining R(t) is q =

 θ θ cos , sin u (with u(t) = 1). 2 2

˙ ω = θ˙ u + sin θ u˙ + (1 − cos θ) u ∧ u.

[1.42]

˙ + sin θ u ¨ ω˙ = θ¨ u + θ˙ ((1 + cos θ) u˙ + sin θ u ∧ u) ¨. +(1 − cos θ) u ∧ u

[1.43]

˙ E XERCISE 1.16.– Express the angular velocities ω and Ω defined in[1.14] with (q, q) ˙ is a consequence and the Maurer-Cartan forms of U. Prove that relation ω˙ = R(q) Ω of the general Proposition 1.1.

1.4. Cayley transforms 1.4.1. Cayley transform defined on La (E) Let O be the complement of the set of U-turns in SO(E): O = {R ∈ SO(E) | −1 ∈ / Spectrum(R)} = {R ∈ SO(E) | R + 1 ∈ Gl(E)}. T HEOREM 1.6.– The maps A → R = (1 + A) ◦ (1 − A)−1 = C(A) −1

R → A = (R − 1) ◦ (R + 1)

=C

−1

(R)

[1.44] [1.45]

are a bijection from La (E) onto O and its reciprocal bijection. Moreover Axis of C(A) = ker(A), C(A) = 1 ⇐⇒ A = 0, C(−A) = C(A)−1 . D EFINITION 1.4.– The map C : La (E) → SO(E) defined in Theorem 1.6 is the Cayley transform. From the standpoint of differential geometry the map [1.45] of an open subset of SO(E) onto an open subset of a vector space La (E) is a chart of the

26

Multi-Body Kinematics and Dynamics with Lie Groups

manifold SO(E). Simple transformations lead to other expressions of the Cayley transform and its reciprocal: C(A) = 2(1 − A)−1 − 1 = (1 − A)−1 ◦ (1 + A), C −1 (R) = 1 − 2(R + 1)−1 = (R + 1)−1 ◦ (R − 1). 2 When A ∈ La (E), the kernel of 1 − A is {0} (for (1 − A)(x) = 0 implies x · x = x · A(x) = 0 and x = 0). Since dim E is finite 1 − A ∈ Gl(E) and the transform is well defined on the whole of La (E). Moreover, if R = C(A): R∗ ◦ R = (1 − A∗ )−1 ◦ (1 + A∗ ) ◦ (1 + A) ◦ (1 − A)−1 = (1 + A)−1 ◦ (1 − A) ◦ (1 + A) ◦ (1 − A)−1 = 1 because all the factors are commuting. We have (1+A)∗ = 1−A hence det(1+A) = det(1 − A) and det C(A) = 1 for A ∈ La (E). After all R ∈ SO(E). That −1 will not be in the spectrum of C(A) = R follows from relation R + 1 = 2(1 − A) where the right hand side is always an invertible operator. When R ∈ O, the relation R = C(A) is equivalent to R − 1 = A ◦ (R + 1) and to A = (R − 1) ◦ (R + 1)−1 . Then (R∗ + 1) = R∗ ◦ (1 + R),

R∗ − 1 = R∗ ◦ (1 − R),

A∗ = (R∗ + 1)−1 ◦ (R∗ − 1) = (1 + R)−1 ◦ (R∗ )−1 ◦ R∗ ◦ (1 − R) = −A. In the conditions of [1.44] properties A(x) = 0 and R(x) = x are equivalent, what is proving the property of the axis of R = C(A). The last two properties are evident. 1.4.2. Cayley transform defined on E ∼

In dimension 3, since La (E)  E by isomorphism a → a, we may express the skew-symmetric operators A of section 1.4.1 with vectors and obtain a map defined on E.

The Displacement Group as a Lie Group

27

T HEOREM 1.7.– Assume that dim E = 3. ∼

i) If A = a then C(A)(x) =

  1 (1 − a2 )x + 2a ∧ x + 2(a · x)a for x ∈ E. 2 1+a

In other words C(A) =

  1 ∼ (1 − a2 )1 + 2 a +2 a ⊗ a . 2 1+a

(where a ⊗ a is the operator such that (a ⊗ a )(x) = (a · x)a for x ∈ E). ii) If A = tan R(u, θ).

θ θ ∼ u (that is to say a = tan u) with u = 1, then C(A) = 2 2

2 Let us express (1 − A)−1 in proving: (1 − A)−1 : x →

1 [x + a ∧ x + (a · x)a] . 1 + a2

[1.46]

To prove [1.46] is equivalent to find y verifying (1 − A)(y) = x or, what is the same, y −a∧y = x. Taking the dot product and the cross product of both sides with a we obtain a · x = a · y and (using Gibbs formula for the double vector product): (1 + a2 )y = x + a ∧ x + (a · x)a. Conversely if y is given by this equation an easy calculation leads to (1 + a2 )(1 − A)(y) = (1 + a2 )x and (1 − A)(y) = x. Relation [1.46] being proved, we have (1 + A) ◦ (1 − A)−1 (x) = (1 + a2 )−1 [x + a ∧ x + (a · x)a +a ∧ (x + a ∧ x + (a · x)a)]  = (1 + a2 )−1 x + 2a ∧ x + 2(a · x)a − a2 x)  = (1 + a2 )−1 (1 − a2 )x + 2a ∧ x + 2(a · x)a , and i) is proved. Under the assumptions of ii): 1 − a2 = cos θ, 1 + a2

2a ∧ x = sin θ u ∧ x, 1 + a2

2(a · x) a = (1 − cos θ)(u · x)u, 1 + a2

and relation C(A) = R(u, θ) follows from Olinde Rodrigues formula [1.18].

28

Multi-Body Kinematics and Dynamics with Lie Groups

1.4.3. Relation between Cayley transform and quaternions ∼

P ROPOSITION 1.4.– When A = a there is one and only one normalized quaternion q ∈ N 4 such that C(A) = R(q) and it is: q= √

1 (1, a). 1 + a2

The so defined relation between a (resp. A) and q is an homeomorphism between E (resp. La (E)) and N . The reciprocal homeomorphism is q = (s, v) → a = s−1 v. 2 By Theorem 1.5 and because C(A) ∈ O, there is one and only one quaternion in N such that C(A) = R(q). If we put q = (s, v) with s= √

1 , 1 + a2

v= √

1 a 1 + a2

then q = (s, v) ∈ N and the formula of Theorem 1.7 is identical to that of Proposition 1.3 [1.35]. When a = tan

θ u, −π < θ < π and u = 1 we find again the quaternion 2

 θ θ  q = cos , sin u corresponding to R = R(u, θ). 2 2 ∼

∼

P ROPOSITION 1.5.– Let A = a, A = a  . i) In order that C(A) ◦ C(A ) ∈ O it is necessary and sufficient that a · a < 1. ∼ 

ii) When this property is verified, then C(A) ◦ C(A ) = C(A ) with A = a a =

and

1 (a + a + a ∧ a ). 1 − a · a

2 Quaternions in N associate to C(A), C(A ) by Proposition 1.4 and one of the quaternions associate to C(A) ◦ C(A ) are: q= √

1 (1, a), 1 + a2

q = √

1 (1, a ), 1 + a2

4 See section 1.3.2 for the definition of N .

q = q.q .

The Displacement Group as a Lie Group

29

Since the real part of q.q is of the form C(1 − a · a ) with C > 0 property i) is clear. Now, if a is defined as in the proposition q.q = √ = √

1 1 √ (1 − a · a , a + a + a ∧ a ) 2 1+a 1 + a2 1 − a · a √ (1, a ). 1 + a2 1 + a2

We have (1 − a · a )2 (1 + a”2 ) = (1 − a · a )2 + (a + a + a ∧ a )2 = 1 − 2a · a + (a · a )2 + a2 + a2 + 2a · a + (a ∧ a )2 = 1 + (a · a )2 + a2 + a2 + a2 a2 − (a · a )2 = (1 + a2 )(1 + a2 ). Hence √

1 − a · a 1 √ =√ , 2 2 1+a 1+a 1 + a2

so that we verify that q.q takes the form mentionned in Proposition 1.4 and ii) is proved. 1.4.4. Angular velocity representation

of

a

motion

described

with

a

Cayley

E XERCISE 1.17.– Prove that, on the open set O, with the parametric representation A → C(A) the right and left Maurer-Cartan forms of G = SO(E) are expressed by: (A, δA) → 2(1 − A)−1 ◦ δA ◦ (1 + A)−1 , (A, δA) → 2(1 + A)−1 ◦ δA ◦ (1 − A)−1 , (the reader who knows more differential calculus in manifolds may remark that these forms are the pull-back of ϑr and ϑ to the manifold La (E) by the map A → C(A)). E XERCISE 1.18.– 1) Prove directly that if a map t → R(t) = C(A(t)) (where the map t → A(t) is derivable) describes a motion of rotation, then the angular velocity is expressed by: ∼ ˙ ◦ (1 + A)−1 , ω = 2(1 − A)−1 ◦ A

d ∼ ∼ ∼ ¨ ◦ (1 + A)−1 + ω ω = 2(1 − A)−1 ◦ A ◦A◦ ω . dt

[1.47] [1.48]

30

Multi-Body Kinematics and Dynamics with Lie Groups ∼

2) Use Maurer-Cartan forms to prove [1.47] and the similar result for Ω (see section 1.2.2). ∼

3) Prove that if A(t) = a (t), then [1.47] and [1.48] read: 2 ˙ (a˙ + a ∧ a), 1 + a2 2 ¨ − 2(a · a)ω). ˙ ω˙ = (¨ a+a∧a 1 + a2 ω=

∼

∼

[1.49] [1.50] ∼

∼

H INT.– Check that (1+ a +a ⊗ a )◦ c ◦(1− a +a ⊗ a ) is the sum of c and four ∼ ∼ groups of terms which, using relations (a ⊗ a )◦ a= 0, a ◦(a ⊗ a ) = 0 and Gibbs formula for the double vector product, may be transformed as ∼

∼

∼

∼

a ◦ c − c ◦ a = (a ∧ c)∼ , ∼ ∼ ∼ ∼ ∼ ∼ (a ⊗ a )◦ c + c ◦(a ⊗ a )− a ◦ c ◦ a = a2 c, ∼ ∼ ∼ ∼ a ◦ c ◦(a ⊗ a ) − (a ⊗ a )◦ c ◦ a = a2 (a ∧ c)∼ , ∼ (a ⊗ a )◦ c ◦(a ⊗ a ) = 0. 1.5. The displacement group as a Lie group In this section we shall study the affine group and the displacement group as Lie groups through their representations as matrix groups and as affine maps and we show that the properties of the differential calculus in these groups are similar to those we have found in sections 1.2 and 1.3 for SO(E) and U. In both forms, the displacement group could be noted D or D(E). Nevertheless we shall note G the matrix form in section 1.5.1 and D the affine form in section 1.5.2 and in the following of the book. It is under the matrix form (or with its representation by 4 × 4) matrices that the Lie group structure of the Euclidean displacement group is generally called up in literature (see for instance [BOY 11] and [BOY 12]). With the terminology of section 1.6 this form is level two whereas the most compact and efficient form for calculations relies on tools of level one. 1.5.1. The displacement group as a matrix group First of all, if we consider a n-dimension real vector space E, the matrices of the form ⎡ ⎤ AU ⎦ with A ∈ L(E), U ∈ E and χ ∈ R M =⎣ 0 χ

The Displacement Group as a Lie Group

31

make a subalgebra A (in particular a linear subspace) of the algebra L(E×R) of linear operators in the vector space E × R. In this algebra the multiplication rule and the unit are ⎤ ⎡ A1 .A2 χ2 U1 + A1 U2 ⎦ M1 M2 = ⎣ [1.51] 0 χ 1 χ2 ⎡ e=⎣

10

⎤ ⎦

(1 = identity operator of E).

01 The above formulae may be understood in two senses leading to equivalent results. In the first sense we consider a n-dimension vector space E and assume that A ∈ L(E) and U ∈ E; then the matrices belonging to A are matrices of operators and no choice of coordinates in E is assumed. In the second sense they are considered as (n + 1) × (n + 1) real matrices expressed as block matrices; in particular A and U are a n × n and a n × 1 real matrices (expressing the operators in some basis of E). The invertible elements of the algebra A make a group A∗ and it is readily verified that: M ∈ A∗ ⇐⇒ A ∈ Gl(E) and χ ∈ R∗ . In A∗ the composition rule is induce by [1.51], the unit element is e and the inverse are expressed as ⎡ M −1 = ⎣

A−1 −A−1 U 0

χ

−1

⎤ ⎦

for M ∈ A∗ .

[1.52]

As it will be explained in section 1.5.2, the matrices M ∈ A∗ with χ = 1 represent the affine transformations of an affine space E into itself, assuming that an origin o or a frame has been chosen in E (the changes of origin on the matrices are governed by [1.78] below). From now on we assume that n = 3, that E is an Euclidean oriented threedimension vector space and that E is an Euclidean affine space over E. The displacements of E are the invertible affine transformations of E such that R ∈ SO(E) and they are described by the matrices ⎡ g=⎣

RU 0 1

⎤ ⎦

with R ∈ SO(E) and U ∈ E.

[1.53]

32

Multi-Body Kinematics and Dynamics with Lie Groups

These matrices make a subgroup of A∗ which will be denoted by G in this section (as we shall explain later, this group is isomorphic to the group D studied in section 1.5.2). In this group, product and inversion are defined, for g, g1 , g2 ∈ G, by: ⎡ g1 g2 = ⎣

R1 R 2 U 1 + R 1 U 2 0

1 ⎡

The unity is e = ⎣

10

⎤ ⎦,

⎡ g −1 = ⎣

R−1 −R−1 U 0

⎤ ⎦.

[1.54]

1

⎤ ⎦ (where 1 is the 3 × 3 identity matrix).

01 The previous formulae may again be understood in a first sense where the matrices belonging to G are matrices of operators and, in a second sense, they are block matrices where R and U are a special orthogonal 3 × 3 matrix and a 3 × 1 matrix (expressing the preceding operators in an orthonormal basis of E; then G appears as a group of 4 × 4 real matrices). The group G is a submanifold of the vector space A. It suffices to remark that A∗ is an open subset of A and G ⊂ A∗ is the inverse image of the point (1, 1) by the differentiable map A∗  M → (R∗ R, χ) ∈ L(E) × R. Since one may verify that this map has a constant rank the conclusion follows from a standard result in differential geometry. Moreover the maps (g1 , g2 ) → g1 g2 and g → g −1 , which are induced by differentiable maps defined on the open subset A∗ , are also differentiable as maps defined on the submanifold G (in other words G is a Lie group). First of all, we shall determine the tangent spaces of the submanifold G of A as a subset of A × A. The result for the displacement group G is linked to the result for the derivative of the maps t → R(t) taking their values in the group SO(E) recalled in section 1.2.2. i) The tangent vector space Te G of the manifold   G at e is the ˜ v ω set of the elements u = (e, u) where u is a matrix of the form with ω and 0 0 v ∈ E, P ROPOSITION 1.6.–

ii) The tangent vector space Tg G of G at g is made of the vectors of the form v = (g, ξ) with: ⎡ ξ=⎣

˜ ω.R v 0

0

⎤

⎡

⎦≡⎣

˜ v R.Ω 0

0

⎤

⎡

⎦ with ω, Ω ∈ E, v ∈ E, g = ⎣

RU 0 1

⎤ ⎦ . [1.55]

The Displacement Group as a Lie Group

33

(Te G and Tg G themselves will often be identified with those vector spaces of matrix by the bijections u → (e, u) and ξ → (g, ξ)). 2 According to a standard definition in differential geometry Te G is the set of the tangent vectors at e of the curves through e lying on G. Assume that a differentiable function t → γ(t) from R to G, with γ(0) = e, defines such a curve, so  that we have  R(t) U(t) two differentiable maps t → R(t) and t → U(t) such that γ(t) = , 0 1 with R(0) = 1 and U(0) = 0. Then ⎡ ⎤ ˜ v ω dγ dU dR ⎦ with ω ˜ = u= (0) = ⎣ (0) and (0) = v. dt dt dt 0 0

[1.56]

˜ and v ∈ E are given, taking R(t) = exp ωt, ˜ where exp is the Conversely, if ω exponential map of SO(E), and U(t) = vt we obtain a curve in G verifying [1.56]. In fact, we have proved i). 

 RU of G. 0 1 Such a vector is the tangent vector at g of some differentiable curve t → γ(t) such that γ(t) ∈ G and γ(0) = g; let us define the curve by We now aim at seeing what is a tangent vector at any point g =

⎡ γ(t) = ⎣

Φ(t) υ(t) 0

⎤ ⎦ with Φ(0) = R, υ(0) = U.

[1.57]

1

dΦ ˜ = R.Ω ˜ ˜ ˜ The derivative, (0) may take two forms ω.Φ(0) = ω.R or Φ(0).Ω dt dυ (0) is a vector v ∈ E. We deduce that all the tangent vectors of G at g are of and dt the form mentioned in [1.55]. The commutator [u1 , u2 ] = u1 u2 − u2 u1 of two matrices of Te G is also an element of Te G since: ⎡ [u1 , u2 ] = ⎣

(ω 1 ∧ ω 2 )∼ ω 1 ∧ v2 − ω 2 ∧ v1 0

⎤ ⎦

[1.58]

0

and it is easy to verify the Jacobi identity [[u1 , u2 ], u3 ] + [[u2 , u3 ], u1 ] + [[u3 , u1 ], u2 ] = 0. Therefore Te G makes a Lie algebra over R.

34

Multi-Body Kinematics and Dynamics with Lie Groups

D EFINITION 1.5.– Endowed with the Lie bracket [1.58], the vector space Te G is the Lie algebra of thegroup G and will be denoted by g (and often identified with the ˜ v ω space of matrices ). 0 0 We may define the left and right translations by a fixed element go of G as Lgo : g → go g, Rgo : g → ggo . These translations are differentiable maps from G to G and we now calculate their tangent maps. As in the proof of Proposition 1.6, we have to calculate LTgo (v) and RgTo (v), where v = (g, ξ) ∈ Tg G is defined by [1.55] and   R o Uo a derivation of [1.57]. For example, if go = : 0 1 ⎡ go .γ(t) = ⎣

⎡ γ(t).go = ⎣

R o Uo 0

⎤ ⎡ ⎦·⎣

1

Φ(t) υ(t) 0

Φ(t) υ(t) 0

⎤ ⎡ ⎦·⎣

⎡

⎦=⎣

Ro .Φ(t) Ro (υ(t)) + Uo 0

1

Ro U o

1

⎤

0

⎤

⎡

⎦=⎣

⎤ ⎦

1

Φ(t).Ro Φ(t)(Uo ) + υ(t) 0

1

⎤ ⎦

1

Taking the derivative we obtain   d  d Lgo γ(t) t=0 = (go .γ(t))t=0 dt dt ⎤⎞ ⎛ ⎡ ˜ Ro (v) Ro .ω.R ⎦⎠ , LTgo (v) = ⎝go g , ⎣ 0 0 ⎤⎞ ⎛ ⎡ ˜ ˜ ω.R.R o v + ω.R(U o) ⎦⎠ RgTo (v) = ⎝ggo , ⎣ 0 0

LTgo (v) =



 ∈ Tg o g G ,

The expanded forms of LTgo (v) = (go g, go .ξ), RgTo (v) = (ggo , ξ.go ). For further calculations note that ⎛

⎡

RgT−1 (v) = ⎝ggo−1 , ⎣ o

−1 −1 ˜ ˜ ω.R.R o v − ω.R.R o (Uo )

0

0

⎤⎞ ⎦⎠ .

The Displacement Group as a Lie Group

35



 ˜ v ω Now we take g = e, v = u ∈ Te G defined by u = and change the notation 0 0   RU go into g defined by g = . We then conclude that, for all g ∈ G and u ∈ g: 0 1 ⎛ ⎡ ⎛ ⎡ ⎤⎞ ⎤⎞ ˜ R(v) ˜ R.ω ω.R ω∧U+v ⎦⎠ , RgT (u) = ⎝g , ⎣ ⎦⎠ , LTg (u) = ⎝g , ⎣ 0 0 0 0 and, compared with [1.55], we may verify that LTg and RgT induce two linear isomorphisms from Te G onto Tg G the reciprocal of which are LTg−1 and RgT−1 . We also note the relations Lg1 ◦Lg2 = Lg1 .g2 , Rg1 ◦Rg2 = Rg2 .g1 and Lg1 ◦Rg2 = Rg2 ◦ Lg1 . Taking the tangent maps and using the chain rule, they imply LTg1 ◦ LTg2 = LTg1 .g2 ,

RgT1 ◦ RgT2 = RgT2 .g1 ,

LTg1 ◦ RgT2 = RgT2 ◦ LTg1 . [1.59]

(It would also be easy to verify [1.59] on the explicit form of LTg1 and LTg2 .) Now we are able to calculate the tangent maps of the fundamental operations of the group π : G × G → G : (g1 , g2 ) → g1 g2 and ι : G → G : g → g −1 . The tangent space of G × G may be identified with the product T G × T G such that, in particular, T(g1 ,g2 ) G×G is identified with the product of vector spaces Tg1 G× Tg2 G. Then: π T (v1 , v2 ) = RgT2 (v1 ) + LTg1 (v2 ) ιT (v) = −LTg−1 RgT−1 (v)

for v1 ∈ Tg1 G, v2 ∈ Tg2 G

for v ∈ Tg G.

[1.60] [1.61]

In order to prove [1.60] it is sufficient to show that π T (v1 , v2 ) = (g1 g2 , ξ 1 g2 + g1 ξ2 ),

if v1 = (g1 , ξ1 ), v2 = (g2 , ξ2 )

However, if we consider two differentiable curves t → γ1 (t) and t → γ2 (t) defining v1 and v2 according to [1.55] and [1.57] the result reduces to the obvious equality:   d = ξ 1 g 2 + g1 ξ 2 . γ1 (t)γ2 (t) dt t=0 In order to prove [1.61] let us take the tangent map of the map g → π(g, ι(g)) = e using [1.60] and solve with respect to ιT (v) the obtained equality:   π T (v, ιT (v)) = RgT−1 (v) + LTg ιT (v) = 0.

36

Multi-Body Kinematics and Dynamics with Lie Groups

1.5.1.1. Adjoint representation of G in g Consider the conjugation h → g.h.g −1 ; the value on u ∈ Te G of its tangent map at h = e will be denoted by Ad g.u. According to the chain rule for the derivative of a composite map Ad g.u = LTg ◦ RgT−1 (u) so that we should obtain a linear representation of G in Te G  g. Indeed, with [1.59], it is easy to verify that g → Ad g is a linear representation, namely that for u ∈ g, g1 and g2 ∈ G: ⎧ ⎨ Ad g1 .(Ad g2 .u) = Ad (g1 g2 ).u ⎩

Ad e = identity of Te G

This way to define the adjoint representation is in agreement with the general theory of Lie groups. However it is often more convenient to work with a representation in the space of matrices isomorphic with Te G and, keeping the same notation, to define Ad g ∈ L(g) by LTg ◦ RgT−1 (u) = (e, Ad g.u)

if u = (e, u)

Performing the calculations we deduce that Ad g.u = g.u.g −1 (a product of matrices) and: ⎡ Ad g.u = ⎣

R(ω)∼ R(v) + U ∧ R(ω) 0

⎤

⎡

⎦ if g = ⎣

0

RU 0 1

⎤

⎡

⎦, u = ⎣

˜ v ω

⎤ ⎦

[1.62]

0 0

However, with [1.62], we may prove more: Ad g.[u1 , u2 ] = [Ad g.u1 , Ad g.u2 ],

[1.63]

so that, Ad g is an automorphism of g and we have a representation of D in its Lie algebra g. 2 On the one hand, taking [1.58] and [1.62] into account, the value of the left side of [1.63] is: Ad g.[u1 , u2 ] ⎤ ⎡ ∼ R(ω 1 ∧ v2 − ω 2 ∧ v1 ) + U ∧ R(ω 1 ∧ ω 2 ) R(ω 1 ∧ ω 2 ) ⎦ =⎣ 0 0

(∗)

The Displacement Group as a Lie Group

37

On the other hand, in the commutator of the matrices on the right side, the first product reads ⎡ (Ad g.u1 )(Ad g.u2 ) = ⎣ ⎡ ⎣

R(ω 1 )∼ R(v1 ) + U ∧ R(ω 1 ) 0

0

R(ω 2 )∼ R(v2 ) + U ∧ R(ω 2 ) 0

⎤ ⎦ ⎤ ⎦

0

 ⎤ R(ω 1 )∼ R(ω 2 )∼ R(ω 1 ) ∧ R(v2 ) + R(ω 1 ) ∧ U2 ∧ R(ω 2 ) ⎦ =⎣ 0 0 ⎡

But, using the properties [1.3] and [1.5], [1.6], the terms of the first row of the commutator are:  ∼ R(ω 1 )∼ R(ω 2 )∼ − R(ω 2 )∼ R(ω 1 )∼ = R(ω 1 ) ∧ R(ω 2 )  ∼ = R(ω 1 ∧ ω 2 )   R(ω 1 ) ∧ R(v2 ) + R(ω 1 ) ∧ U ∧ R(ω 2 ) − R(ω 2 ) ∧ R(v1 )   −R(ω 2 ) ∧ U ∧ R(ω 1 )   = R(ω 1 ∧ v2 − ω 2 ∧ v1 ) + R(ω 1 ) ∧ U ∧ R(ω 2 )   −R(ω 2 ) ∧ U ∧ R(ω 1 )   = R(ω 1 ∧ v2 − ω 2 ∧ v1 ) + U ∧ R(ω 1 ) ∧ R(ω 2 ) = R(ω 1 ∧ v2 − ω 2 ∧ v1 ) + U ∧ R(ω 1 ∧ ω 2 ) so that we find the expression (∗) again and [1.63] is proved. 1.5.1.2. Maurer-Cartan forms The way to define the Maurer-Cartn forms in the general theory of Lie groups is the following: if v ∈ Tg G the elements LTg−1 (v) and RgT−1 (v) belong to Te G so that we can define two maps ϑ and ϑr : T G → Te G such that: ϑ (v) = LTg−1 (v),

ϑr (v) = RgT−1 (v) when v ∈ Tg G.

[1.64]

For a fixed g, ϑ and ϑr induce linear maps from Tg G to g; they define “differential forms” on G with values in g named left and right Maurer-Cartan forms. Let us turn to the description of ϑ and ϑr when G is the displacement group. If v = (g, ξ) is

38

Multi-Body Kinematics and Dynamics with Lie Groups

defined as in [1.55], strictly speaking LTg−1 (v) = (e, g −1 .ξ), RgT−1 (v) = (e, ξ.g −1 ). When Te G is identified with the space of matrices g, by simple calculations: ⎡∼ ⎤ −1 ⎢ Ω R (v) ⎥ ϑ (v) = ⎣ ⎦, 0 0

⎡ ϑr (v) = ⎣

˜ v+U∧ω ω 0

⎤ ⎦

[1.65]

0

(with v = (g, ξ) defined according to [1.55]). Maurer-Cartan forms are related by: ϑr (v) = Ad g.ϑ (v)

if g is the origin of v.

[1.66] ∼

˜ = (R−1 (ω))∼ = Ω, 2 Taking account of relation Ω = R−1 .ω and using R−1 .ω.R the result in a straightforward consequence of [1.62] where ω is replaced by Ω and v by R−1 .v and of [1.65]. A more general proof of [1.55] could also be deduced from [1.59]. R EMARK 1.1.– In [1.60] π T (v1 , v2 ) ∈ Tg1 .g2 G so that, according to the definition of Maurer-Cartan forms:   ϑr π T (v1 , v2 ) = ϑr (v1 ) + Ad g1 .ϑr (v2 ),   ϑ π T (v1 , v2 ) = Ad g2−1 .ϑ (v1 ) + ϑ (v2 ). Regarding the inverse in G, if v ∈ Tg G then ιT (v) ∈ Tg−1 G so that, according to [1.61]:   ϑ ιT (v) = −ϑr (v),

  ϑr ιT (v) = −ϑ (v).

These relations prove that Maurer-Cartan forms are exchanged each other by the inverse ι in G and the opposite in g. The previous formulas are important in kinematics of compounded motions. 1.5.1.3. Differential of the adjoint representation The map g → Ad g is a map from G to the vector space L(g) of the endomorphisms of the vector space g. Its differential is a map from T G to L(L(g)) transforming linearly each vector space Tg G into (linear) operators in g. In other words, for v ∈ Tg G, DAd (v) ∈ L(g): P ROPOSITION 1.7 (Differential of the adjoint representation of G).– The differential of the map g → Ad g from D to L(D) may be expressed by two ways: DAd (v) = ad ϑr (v) ◦ Ad g = Ad g ◦ ad ϑ (v) for v ∈ Tg G

[1.67]

The Displacement Group as a Lie Group

39

or, what is equivalent: DAd (v).uo = [ϑr (v), Ad g.uo ] = Ad g.[ϑ (v), uo ]

[1.68]

(This result for the displacement group is formally similar to Proposition 1.1 for the orthogonal groups.) 

 ˜ o vo ω 2 Assume that v is defined according to [1.57] and [1.55] and let uo = be a 0 0 fixed element of g, then:  DAd (v).uo =

d Ad γ(t).uo dt

 t=0

However ⎡ Ad γ(t).uo = ⎣

Φ(t)(ω o )∼ Φ(t)(vo ) + υ(t) ∧ Φ(t)(ω o ) 0

⎤ ⎦

0

hence: DAd (v).uo ⎤ ⎡ ∼     ω ∧ R(ω o ) ω ∧ R vo + U ∧ ω ∧ R(ω o ) + v ∧ R(ω o ) ⎦ =⎣ 0 0 ∼  The appearance of ω ∧ R(ω o ) suggests to search for another expression with a Lie bracket. In fact, ⎡ ϑr (v) = ⎣

˜ v+U∧ω ω 0

0

⎤ ⎦,

⎡ Ad g.uo = ⎣

R(ω o )∼ R(vo ) + U ∧ R(ω o ) 0

⎤ ⎦

0

and, with [1.58], we find [ϑr (v), Ad g.uo ]   ⎤ ⎡ (ω ∧ R(ω o ))∼ ω ∧ R(vo ) + U ∧ R(ω o ) − R(ω o )(v + U ∧ ω) ⎦ =⎣ 0 0

40

Multi-Body Kinematics and Dynamics with Lie Groups

Jacobi identity for the cross product gives ω ∧ (U ∧ R(ω o )) − R(ω o ) ∧ (U ∧ ω) = U ∧ (ω ∧ R(ω o )) and the transformation of the second term of the first row of the matrix leads to: dAd (v).uo = [ϑr (v), Ad g.uo ] Making use of [1.63] and [1.66] we find the analogous expression with ϑ (which could also be verified by direct calculations): dAd (v).uo = Ad g.[ϑ (v), uo ] 1.5.1.4. Complement: Maurer-Cartan formulae Maurer-Cartan formulae express the exterior differentials of the Maurer-Cartan forms. They are verified in every Lie group, however the proofs given below are specific of the matrix groups setting. Those formulas are important in differential calculus because they provide us with the rule for exchanging the order of partial derivatives for mapping with range in a Lie group; a rule which is not so simple as Schwarz theorem for mappings with range R. There is a slight difficulty because we have to operate with g-valued differential forms. Nevertheless, if we consider that such a form is determined by its components in a basis of g, which are ordinary differential forms, then the exterior differential may be clearly defined component by component. It may be checked that the result is independent of the choice of the basis and that the standard rules of exterior calculus extend to g-valued forms in a natural way provided that multiplication would be understood as multiplication of matrices. With matrices, Maurer-Cartan forms may be expressed by ϑr = dg.g −1 and ϑ = g −1 .dg where the dot means the product of matrices, and when v = (g, ξ) ∈ Tg G is defined according to [1.55]: dg(v) = ξ,

ϑr (v) = ξ.g −1 ,

dϑr = −dg ∧ d(g −1 ),

ϑ (v) = g −1 .ξ,

dϑ = d(g −1 ) ∧ dg,

where d(g −1 ) = −g −1 .dg.g −1 is the ordinary differential of the map g → g −1 (remember that, for exterior differential forms α and β, d(α ∧ β) = (dα) ∧ β + (−1)p α ∧ dβ where p is the degree of α) so that, if v1 = (g, ξ 1 ) and v2 = (g, ξ2 ) are two elements of Tg G then: dϑr (v1 , v2 ) = dg(v1 ).g −1 .dg(v2 ).g −1 − dg(v2 ).g −1 .dg(v1 ).g −1 = ξ 1 .g −1 .ξ 2 .g −1 − ξ 2 .g −1 .ξ 1 .g −1 = [ϑr (v1 ), ϑr (v2 )] dϑ (v1 , v2 ) = −g −1 .dg(v1 ).g −1 .dg(v2 ) − (−g −1 .dg(v2 ).g −1 .dg(v1 )) = g −1 .ξ 2 .g −1 .ξ 1 − g −1 .ξ 1 .g −1 .ξ 2 = −[ϑ (v1 ), ϑ (v2 )]

The Displacement Group as a Lie Group

41

Finally we have proved the two formulas: dϑr = [ϑr , ϑr ],

dϑ = −[ϑ , ϑ ]

[1.69]

Here is the result for exchanging the order of partial derivatives: P ROPOSITION 1.8.– Let U be an open subset of Rn (n integer ≥ 2) and f : U → G be a C 2 mapping. Put, for i, j = 1, . . . , n: 

 ∂f (q) ∂qi   ∂f −1 ∂f Wi (q) = f (q) . (q) = ϑ (q) ∂qi ∂qi Vi (q) =

∂f (q).f (q)−1 = ϑr ∂qi

(right partial derivative), (left partial derivative).

Then:   ∂Vj ∂Vi − = V i , Vj , ∂qi ∂qj

  ∂Wi ∂Wj − = Wj , W i . ∂qi ∂qj

2 The proof could be deduced from [1.69]. However we shall write an independent proof on the example of the left-derivatives: ∂Wj ∂ (q) = ∂qi ∂qi

 f (q)

−1

∂f . (q) ∂qj



∂f ∂f ∂2f (q).f (q)−1 . (q) + (q) ∂qi ∂qj ∂qi ∂qj  ∂f   ∂f  ∂2f (q) .ϑ (q) + (q) = −ϑ ∂qi ∂qj ∂qi ∂qj  ∂f   ∂f  ∂Wi ∂2f (q) = −ϑ (q) .ϑ (q) + (q). ∂qj ∂qj ∂qi ∂qj ∂qi = −f (q)−1 .

The calculations are correct because it is considered that the mapping f ranges in a vector space of matrices and in this framework the meaning of its partial derivative is clear. In particular since f is C 2 , Schwarz theorem for the exchange of the order of partial derivative is verified and we find  ∂f  ∂f   ∂f    ∂f  ∂Wj ∂Wi (q) − (q) = ϑ (q) .ϑ (q) − ϑ (q) .ϑ (q) ∂qi ∂qj ∂qj ∂qi ∂qi ∂qj = −[Wi (q), Wj (q)] = [Wj (q), Wi (q)]. (because the Lie bracket is the commutator of matrices.

[1.70]

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Multi-Body Kinematics and Dynamics with Lie Groups

1.5.1.5. Klein form on g The existence of a Lie algebra structure on g and of the adjoint representation of G in g as shown in the preceding sections was a particular case of general properties of Lie groups. We now turn to a specific feature of the Euclidean displacement group. The Klein form is the bilinear form on g defined by ⎡ [[u1 | u2 ]] = ω 1 · v2 + ω 2 · v1 if ui = ⎣

˜ i vi ω 0 0

⎤ ⎦ , i = 1, 2.

[1.71]

It is readily verified that the Klein form is a symmetric non-degenerate bilinear form on g so that it defines a pseudo-Euclidean structure on g and an isomorphism of g onto g∗ : every linear form f ∈ g∗ may be expressed as u → f (u) = [[ϕ | u]] with a uniquely defined ϕ ∈ g. The Klein form is invariant by the adjoint representation: for g∈G [[Ad g.u1 | Ad g.u2 ]] = [[u1 | u2 ]].

[1.72]

2 According to [1.62], taking into account the skew-summetry of the mixed product and the invariance of the dot product under the orthogonal group of E:   [[Ad g.u1 | Ad g.u2 ]] = R(ω 1 ) · R(v2 ) + U ∧ R(ω 2 )   +R(ω 2 ) · R(v1 ) + U ∧ R(ω 1 ) = R(ω 1 ) · R(v2 ) + R(ω 2 ) · R(v1 ) = ω 1 · v2 + ω 2 · v1 = [[u1 | u2 ]],

The relation between the Klein form and the Lie bracket is [[ad u.u1 | u2 ]] + [[u1 | ad u.u2 ]] = 0

[1.73]

This formula means that ad u is a skew-symmetric operator in the pseudoEuclidean space g. 2 Formula [1.73] may be proved by direct calculation. However, it is an infinitesimal form of [1.72] and a better proof consists in evaluating the derivative of [1.72] with [1.68] at g = e on u = (e, u) ∈ Te G.

The Displacement Group as a Lie Group

43

1.5.1.6. The exponential map of the group G The exponential map of a Lie group appears when we study the flow (that is to say the integrals) of a left-invariant vector field on the group. A left-invariant vector field on G is a map X : G → T G of the form X(g) = LTg (u) with u = X(e) ∈ Te G. For the particular group we consider, X(g) = (g, g.u) if u = (e, u) and the problem amounts to the following:   for a given matrix u ∈ g, search a map t → γ(t), defined on an interval I ∈ R with 0 ∈ I, such that: dγ (t) = γ(t).u, t ∈ I, γ(0) = go . dt The maximal solution to this problem is known to be defined on I = R by: γ(t) = go . exp(tu) where exp means the ordinary exponential of matrices and it is sufficient to calculate exp u for u ∈ g. First of all, note the following property which is only the matter of simple calculation:   ˜ v ω L EMMA 1.3.– Consider an element ∈ g with ω = 0. Then: 0 0 ⎡ ⎣

1 −a

⎤ ⎡ ⎦×⎣

0 1

˜ v ω

⎤ ⎡ ⎦×⎣

0 0

1a

⎤

⎡

⎦=⎣

˜ fu ω ω 0

01

 From [1.74] we deduce if u = ⎡ ⎢ exp u = ⎣ ⎡ exp u = ⎣

˜ exp ω 0 1v

0

⎤ ⎦ with a = 1 ω ∧v, fu = ω · v .[1.74] ω2 ω2

 ˜ v ω ∈ g with ω and v ∈ E then 0 0

⎤ 1  ˜ ˜ ω 1 − exp ω) ◦ ω(v) + f u ⎥ ω2 ⎦ if ω = 0,

[1.75]

1

⎤ ⎦ if ω = 0.

[1.76]

01  2 For proving [1.75] and [1.76] it suffices to remark that for n ≥ 2, hence: ⎡ exp ⎣

˜ fu ω ω 0

0

⎤

⎡

⎦=⎣

˜ fu ω exp ω 0

1

⎤ ⎦,

˜ fu ω ω 0 0

n

 =

˜n 0 ω 0 0



44

Multi-Body Kinematics and Dynamics with Lie Groups

and to expand the relation ⎡ exp u = ⎣

1a

⎤

⎡

⎦ × exp ⎣

01

˜ fu ω ω 0

⎤

⎡

⎦×⎣

0

1 −a 0 1

⎤ ⎦ with a = 1 ω ∧ v. ω2

The case ω = 0 is evident. R EMARK 1.2.– When ω = 0the result may be interpreted as follows: if we consider ˜ v ω that the matrix u = describes a “torsor” with moment v at an origin o, 0 0 Lemma 1.3 expresses this torsor when the origin lies on its “axis” where the moment is colinear to ω. The matrix u may also describe a screw with respect to an origin o in the Euclidean affine space (see Theorem 1.12 for the intrisic aspect). Then, with  − → ˜ ω ω f u respect to an origin o such that oo = a, the matrix reads where, now, “the 0 0  vector v” is colinear with ω (and fu is the “pitch” of the screw); o lies on the axis of the screw. Then exp u is an helical displacement about the axis of the screw u, easy to calculate when the origin lies on the axis of u. After all, the calculation of exp u may be reduced to the calculation when the origin lies on the axis followed by a change of origin as explained in [1.78] below. 1.5.2. The displacement group as a group of affine maps Let E be the Euclidean affine space in dimension 3 and E be the associated vector space. From a first point of view, to every pair (a, b) of points is associated −→ a vector a, b ∈ E verifying: ⎧− → − → → ⎪ ac (Chasles’ relation), ⎨ ab + bc = − ⎪ → ⎩− ab = 0 ⇐⇒ a = b. From a second point of view, the additive group of E acts transitively and freely on E: i.e. there is an action (a, x) → a + x such that, for a and b in E, x and y in E: ⎧ ⎨ (a + x) + y = a + (x + y), ⎩

for all a and b ∈ E there is one and only one x ∈ E such that a + x = b,

in particular a + x = a ⇐⇒ x = 0. For fixed x ∈ E, the map a → a + x is the translation defined by x and the additive group of E “is” the translation group of E. − → The link between those two points of view reads a + x = b ⇔ x = ab.

The Displacement Group as a Lie Group

45

An affine transformation of E is a mapping A : E → E such that there exists a linear map A ∈ L(E) (called the linear part of A and denoted by the same letter in boldface) such that the following equivalent properties be verified for all a, b in E and x ∈ E: − −−−−−−→ → A(a)A(b) = A ab ⇐⇒ A(a + x) = A(a) + A(x). The frameworks of affine transformations and the formalism of matrices explained at the head of section 1.5.1 may be translated one into the other according to the easily proved rule: If o is an origin in E and A is an affine transformation of E then its action on the point p may be calculated according to: 

p ∈ E,

→ x=− op,

      −−−→ −−−→ y AU x y = oA(p) ⇐⇒ = . with U = oA(o). 1 0 1 1

Hence there is a bijection  A → M = M(A) =

AU 0 1



between the affine transformations of E and the matrices of section 1.5.1. Moreover, if A1 and A2 are affine transformations of E then A1 ◦A2 is also an affine transformation the linear part of which is A1 ◦ A2 and for the associated matrices M(A1 ◦ A2 ) = M(A1 ) ∧ M(A2 ).   Moreover, A is a bijection from E onto E ⇐⇒ A ∈ Gl(E) and then M A−1 = M(A)−1 . 2 If we put B = A1 ◦ A2 then   → −−−→ oB(p) = U1 + A1 U2 + A1 ◦ A2 − op . Therefore, the associate matrix of B is defined by [1.51] because 

      A1 U 1 A2 U 2 A1 .A2 U1 + A1 U2 = × . 0 1 0 1 0 1

Solving in x the above relation between x and y one find A−1 (as in formula [1.52]):    −1    x A −A−1 U y = · . 1 0 1 1

46

Multi-Body Kinematics and Dynamics with Lie Groups

We first deduce that: The bijective affine transformations make a group Ga(E) which is isomorphic to the matrix group A∗ of section 1.5.1 by the relation A → M (A) = M (when an origin has been chosen in E). R EMARK 1.3.– If o and o are origins in E then, for any point m ∈ E, Chasles’ relation −−→ → → − a with − leads to o m = − om oo = a. Therefore the representations by matrices of E and of the affine transformations with respect to the two origins are related by: ⎡

⎤

⎤ ⎡ ⎤ x −→ → x = − ⎣ ⎦=⎣ ⎦ · ⎣ ⎦ if x = − o m, om, 1 0 1 1 x

⎡

1 −a

⎡ Mo (A) = ⎣

1 −a

⎤

⎡

⎦ · Mo (A) · ⎣

0 1

1a 01

⎤

⎡

⎦ = ⎣

[1.77]

A (A − 1)a + U 0

⎤ ⎦

[1.78]

1

The following results are classical in geometry and in mechanics: i) A ∈ Ga(E) is a displacement (resp. an isometry) of E if and T HEOREM 1.8.– only if A ∈ SO(E) (resp. O(E)). ii) The displacements of E make a subgroup D = D(E) of Ga(E) which is isomorphic to the matrix group of section 1.5.1 (when an origin has been chosen in E). Recall that the isometries of E are the affine mappings preserving the distance and the displacements of E are the isometries of E preserving orientation. E XERCISE 1.19.– ∗ This exercise requires some knowledge of topology. Let G be the  RU matrix group of section 1.5.1 (matrices of the form g = with R ∈ SO(E) 0 1 and U ∈ E). Put δ(g1 , g2 ) = R2 − R1 + |U2 − U1 | fot g1 and g2 ∈ G. 1) Check that δ defines a distance on G and prove that the operations of the group G are continuous. 2) Prove that the subsets A = {g ∈ G | R ∈ SO(E), U = 0},

U = {g ∈ G | R = 1, U ∈ E}

The Displacement Group as a Lie Group

47

are closed subgroups of G, isomorphic with SO(E) and the additive group of E and that A is a compact subset of G. 3) For any choice of the origin o in the affine space E let us define the map do : D× D → R (D = D(E)) such that   do (A2 , A1 ) = δ Mo (A2 ), Mo (A2 ) Check that do is a distance on D, that the associated topology on D is independent of the choice of o and that the operations in the group D are continuous. H INT.– See also exercise 1.4, section 1.2. For 3) see [1.78]. 1.5.3. Classification of the Euclidean displacements T HEOREM 1.9.– Let FA denote the set of fixed points of A ∈ D. There are four classes of displacements characterized by: i) A = 1, FA ii) A = 1, FA iii) A = 1, FA iv) A = 1, FA

=E =∅ =Δ =∅

identity, translations, rotations, helical displacements.

In the case iii) Δ is a straight line (the axis of the rotation). In the case iv) there exists a straight line Δ (the axis of the helical displacement) such that A(Δ) = Δ (Δ is globally invariant). In both cases the direction of Δ is the eigensubspace associate to the eigenvalue 1 of A. −−−→ L EMMA 1.4.– For A ∈ D define the map ΦA : E → E by ΦA (p) = pA(p) for p ∈ E. Then, ΦA is an affine map and its linear part is A−1. If u is a normalized vector such that A(u) = u (eigenvector of A associated to the eigenvalue 1), then the number Φa (p) · u = δA is constant over E. −−−→ 2 Let o be a fixed origin in E, ξ = oA(o) = ΦA (o), we have → −−−→ oA(p) = ξ + A − op), − −−−→ → → pA(p) = ξ + A → op) − − op = ξ + (A − 1)(− op), proving that ΦA is affine with the mentioned linear part. Now, since u = A(u):   → → → ΦA (p) · u = ξ · u + (A − 1).− op · u = ξ · u + A − op · A(u) − − op · u − → − → = ξ · u + op · u − op · u = ξ · u = Φ (o) · u. A

48

Multi-Body Kinematics and Dynamics with Lie Groups

proving that the value of ΦA (p) · u is constant over E.  −1  L EMMA 1.5.– Let T = Tr (−δA u) = Tr (δA u) . Then A ◦ T = T ◦ A. 2 Since A ◦ T and T ◦ A have the same linear part, it is sufficient to prove that the image of o by the two maps is the same. −−−−→ −−−→ −−−−−−−→ −−−→ oT A(o) = oA(o) + A(o)T A(o) = oA(o) − δA u, −−−→ −−−→ −−−−→ −−−→ −−−−−−−→ −−−→ oAT (o) = oA(o) + A(o)AT (o) = oA(o) + A oT (o) = oA(o) − δA u, −−−→ because oT (o) = −δA u is an eigenvector of A associated to the eigenvalue 1. 2 Let us turn to the proof of the theorem. When A = 1 the map ΦA is a constant −−−→ −−−→ → vector over E. If this constant is a vector ξ ∈ E we have pA(p) = oA(p) − − op = ξ, therefore A(p) = p + ξ for all p ∈ E, hence A = Tr (ξ) and we are in case i) or ii) according to ξ = 0 or ξ = 0. When A = 1, let u be a normalized vector in the kernel of A so that the range of A − 1 is (Ru)⊥ and its kernel is Ru. The fixed points of A are determined by: → c is a fixed point of A ⇐⇒ (A − 1)(− oc) = −ξ

()

therefore we have two and only two cases: a) ξ ∈ (Ru)⊥ (i.e. δA = ξ·u = 0). Then there exists co such that () is verified and since A−1 induces a linear automorphism of the subspace (Ru)⊥ the point co may be → ·u = 0 and the whole set of fixed points is then Δ = c +Ru = F , chosen so that − oc o o A −−−−−−−→ −−−−→ a straight line by c with director u. For all p ∈ E, relation A(c )A(p) ≡ c A(p) = o o o  → A − co p reads:  → A(p) = co + A − co p . giving the general expression of A. Let Π be the affine plane orthogonal to u by o, since A sends (Ru)⊥ into itself, this expression shows that Δ cuts Π at co , transforms Π into itself and induces a plane rotation with center co on Π: after all, the geometrical meaning of our results is that A is a rotation about the axis Δ. b) ξ ∈ / (Ru)⊥ (i.e. δA = ξ · u = 0). Then FA = ∅.

−−−→ Let us consider the map R = A ◦ T = T ◦ A (see Lemma 1.5). We have oR(o) = − − − → ξ − δA u ∈ (Ru)⊥ , R = A = 1 and a fixed point of R must verify cR(c) = 0 what is equivalent to → (R − 1)(− oc) = −(ξ − δA u)

()

The Displacement Group as a Lie Group

49

Hence, up to the change of notations A and ξ into R and ξ − δA u, we conclude as in case (a) that R is a rotation about an axis Δ, axis directed as u by a solution co of (). Then A = Tr (δA u) ◦ R = R ◦ Tr (δA u) is a rotation about Δ compounded with a translation in the direction of Δ that is to say a helical displacement (or a “screwing” about Δ). In the course of the proof the following result was also proved: T HEOREM 1.10 (Chasles’ Theorem).– If A is neither the identity nor a translation, there exists an axis (Δ, u), a translation T = Tr (δu) in the direction of Δ and a rotation R = R(Δ, θ) (about Δ) such that A = T ◦ R = R ◦ T (T is the identity when A is a rotation). R EMARK 1.4.– More generally: let A be a displacement and T = Tr (x) a translation. In order that A and T commute it is necessary and sufficient that A(x) = x. This property is trivial when A is the identity or a translation (the translation group is a commutative subgroup of D(E)). 1.5.4. The Lie algebra of D as a Lie algebra of vector fields on E T HEOREM 1.11.– Let X : E → E be a vector field on E. The following properties are equivalent a) X is a skew-symmetric vector field, that is to say:   → for all p and q ∈ E : X(p) − X(q) · − pq = 0. b) X is a moment field, that is to say: there is a vector ω such that: → for all p and q ∈ E : X(q) = X(p) + ω ∧ − pq. Then the vector ω is uniquely defined (and it will be denoted by ω X ). The proof of this theorem is the matter of exercise 1.1. Those vector fields make a vector space D(E) over R we shall note simply D in the following; of course addition and product by scalars of elements of D are defined by: (X + Y )(p) = X(p) + Y (p),

(λX)(p) = λX(p) for p ∈ E,

where to the right we have the operations of E. It is evident that the map X → ω X is linear. We shall define T as the kernel of this map, i.e. the linear subspace of D such that ω X = 0 of the constant vector fields on E.

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Multi-Body Kinematics and Dynamics with Lie Groups

It is noteworthy that property a) appears in a natural way in kinematics for the velocity fields of the particles of a rigid body whereas property b) appears in dynamics for the moment fields of “torsors”. When o is fixed in E, an element of D is determined by the pair of vectors ω X , X(o) (Plücker’s vectors defining a “torsor” by its sum and its moment about o when X is regarded as a moment field); the map Jo : X → (ω X , X(o)) is an isomorphism between the vector spaces D and E × E (however this isomorphism depends on the choice of an origin in E and the best way is to operate with D what is free of such a choice...). T HEOREM 1.12 (Axis of an element of D).– Let X ∈ D such that ω X = 0 (i.e. X ∈ / T). Then there exist an axis ΔX in E (straight line with director ω X ) such that the following properties are equivalent: a) p ∈ ΔX , b) ω X ∧ X(p) = 0, c) X(p) = fX ω X with fX =

1 [[X | X]] , 2 ω 2X

d) X(p) = min{ X(a) | a ∈ E}. Theorem 1.12 is an infinitesimal form of Chasles’ Theorem: when X describes the velocity field of the particles of a rigid body, it shows that if ω X = 0, this field is that of a “screwing” motion about ΔX and with pitch 2πfX (and that of a translation motion when ω X = 0). In classical kinematics, an element of D such that ω X = 0 is regarded as a “screw” with “pitch” fX about the axisΔX . Property  (c) means that, at any point of the axis, the Plücker’s vectors of X, are ω X , fX ω X .   origin in E; X is defined by X(o), ω X . Property (b) reads  2 Let o be a fixed → op ∧ ω X = 0 and, using Gibbs formula, becomes: X(o) + ω X ∧ −   → → op ω X = ω 2X − ω X ∧ X(o) + ω X · − op, → an equation where the unknown is − op. A particular solution is the point po such that: − → = 1 ω ∧ X(o) op o ω 2X X → → + λω with λ ∈ R. All these and all the other solutions are of the form − op = − op o X points p make a straight line ΔX by po , directed as ω X and such that (a) ⇔ (b) is verified. When ΔX is so defined, for all p ∈ ΔX , using Gibbs formula, we have  →   1  X(p) = X(o) + ω X ∧ − opo + λω X = 2 ω X · X(o) ω X = fX ω X . ωX

The Displacement Group as a Lie Group

51

Hence (b) ⇔ (c) and (a), (b) and (c) are equivalent. p→ When po ∈ ΔX and p is any point of E we have X(p) = X(po ) + ω X ∧ − op with X(po ) = fX ω X . Since the vectors to the right are orthogonal: X(p) 2 = 2

X(po ) 2 + ω X ∧− p→ o p . Therefore the lower bound of X(p) for p ∈ E is X(po ) and it is reached if and only if ω X ∧ − p→ o p = 0, that is if and only if p ∈ ΔX . Hence (a) ⇔ (d). Let us define A∗ X as the image of X ∈ D under action of A ∈ D(E) according to the standard definition in mathematics, namely the vector field on E such that:   A∗ X(p) = A X(A−1 (p) for p ∈ E.

[1.79]

It is readily proved that Y = A∗ X belongs to D and that   ωY = A ωX ,

    −−−→ Y (o) = A X(o) − A ω X ∧ oA(o)

[1.80]

Let us define the Lie bracket of X and Y ∈ D as the vector field [X, Y ] such that [X, Y ](p) = ω X ∧ Y (p) − ω Y ∧ X(p) for p ∈ E

[1.81]

Then [X, Y ] ∈ D with ω [X,Y ] = ω X ∧ ω Y and this bracket defines a Lie algebra structure on D so that X → ω X is a homomorphism of Lie algebras (see exercise 1.21). Then A∗ [X, Y ] = [A∗ X, A∗ Y ]

[1.82]

Moreover A → A∗ is a linear representation of D in the Lie algebra D (that is to say (AB)∗ = A∗ ◦ B∗ and e∗ = identity). The Lie algebra D contains remakable subspaces: Za = {X | X(a) = 0} for fixed a ∈ E,

T = {X | ω X = 0}.

[1.83]

It is readily proved that [X, Y ] ∈ Za when X, Y ∈ Za and [X, Y ] ∈ T when X ∈ D and Y ∈ T, [X, Y ] = 0 when X and Y ∈ T. Therefore Za is a Lie subalgebra of D (corresponding to the subgroup of rotations about a) and T is a commutative ideal of D (corresponding to the subgroup of translations). P ROPOSITION 1.9.– Let A ∈ D, X ∈ D such that ω X = 0 and Y = A∗ X. Then, as directed line in E, ΔY is the transformed of ΔX by the displacement A.

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P ROPOSITION 1.10.– Let X and Y ∈ D, then (if means parallel.): ⎧ either X and Y are not in T and ΔX = ΔY , ⎪ ⎪ ⎨ or X ∈ / T, Y ∈ T and Y ΔX , [X, Y ] = 0 ⇐⇒ or X ∈ T, Y ∈ / T and X ΔY , ⎪ ⎪ ⎩ or X and Y ∈ T.  [X, Y ] ∈ T ⇐⇒

either X and Y are not in T and ΔX ΔY or (X ∈ T or Y ∈ T)

The proof of Propositions 1.9 and 1.10 are the matter of exercise 1.35 and of a part of exercise 1.36. A general theorem says that when a Lie group G acts on a manifold M by an effective differentiable action, there is an isomorphism between its Lie algebra g and a Lie algebra of vectorfields on M (see [KOB 63] Chap. I, section 4 or [CHE 06] Chap. V, section 2). The following Proposition 1.11 means that, for the Euclidean displacement group acting on the Euclidean space E, the isomorphic Lie algebra is D, in some sense the representation A → A∗ is equivalent to the adjoint representation of D(E) and the Lie bracket in D is the infinitesimal form of the representation A → A∗ . In other words D, appearing in a natural way in kinematics, may be regarded as the Lie algebra of that group. First of all let us specify that a map t → At from an interval I (⊂ R) to D(E) will be said to be differentiable if for all p ∈ E the map t → At (p) is differentiable from I to E. P ROPOSITION 1.11.– Let t → At be a differentiable mapping from R to D(E) such that A0 = e (unity of D(E)=identity). There is a vector field X ∈ D such that:  for all p ∈ E :

d At (p) dt

what will be also written 

 t=0

= X(p)

[1.84]

d At = X. With the so defined X: dt

 d −1 = −X(p) for p ∈ E, At (p) dt t=0   d = [X, Y ] for fixed Y ∈ D, At∗ Y dt t=0

[1.85] [1.86]

The Displacement Group as a Lie Group



 d = D∗ X for fixed D ∈ D(E). IntD.At dt t=0   d −1 = −[X, Y ] for fixed Y ∈ D. At∗ Y dt t=0

53

[1.87] [1.88]

−−−−−−−→ → 2 For all p and q ∈ E, by definition of a displacement At (q)At (p) 2 = − qp 2 . If X is defined by equality [1.84]: 1 2



d −−−−−−−→ 2

At (q)At (p) dt

 t=0

  → = X(p) − X(q) · − pq = 0.

Hence, X ∈ D by Theorem 1.11 a). To find the vector ω X of Theorem 1.11 −−−−−−−→ → b) remark that, by definition of an affine map, At (q)At (p) = At (− qp). Taking the derivative at t = 0:   d ∼ → X(p) − X(q) = ω ∧ − qp, with ω = . At dt t=0  −1 (p) with a fixed p and taking the derivative of relation Now, putting q(t) = At d  At (q(t)) = p leads to dt q(t) t=0 + X(q(0)) = 0, therefore: 

 d −1 = −X(p) for all p ∈ E, A (p) dt t t=0       d d = (p)) At∗ Y (p) At Y (A−1 t dt dt t=0 t=0     dAt d −1 ∼ = (Y (p)) + A0 ◦ ω Y ◦ At (p) dt t=0 dt t=0 = ω X ∧ Y (p) − ω Y ∧ X(p) = [X, Y ](p) for all p ∈ E.

The proof of [1.86] follows from [1.84] taking the derivative     D ◦ At ◦ D−1 (p) = D A0 (D−1 (p) + D (At − A0 )(B −1 (p) . T HEOREM 1.13.– Let A ∈ D(E) and t → A(t) be a differentiable map from an interval I ⊂ R to D(E). Putting A(to ) = A (for a fixed to ∈ I):

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Multi-Body Kinematics and Dynamics with Lie Groups

i) There exists V and W in D related by V = A∗ W and such that, for all p ∈ E: ⎧   ⎨ V (A(p)) d or [1.89] = A(t)(p) ⎩ dt t=to A(W (p)) ii) For all fixed X ∈ D the derivative of the map t → A(t)∗ X valued in D is: ⎧   ⎨ ad V.A∗ X d or = [1.90] A(t)∗ X ⎩  dt t=to A∗ ad W.X) The conclusions of Theorem 1.13 are very important in kinematics and dynamics. When t → At describes the motion of a rigid body the vector field V on E describes the velocity field of the particles of the body in a space fixed frame (Eulerian standpoint) and W describes this velocity field in a body fixed fixed frame (Lagrangian standpoint). In formula [1.89] p may be understood as the position of a particle in a reference position of the body. At time to the present position of this particle in space is then A(p) = Ato (p) and, in formula [1.89], the vector field V acts on this present position whereas the vector field W acts on p. The vector ω V is the angular velocity of the body. 2 The derivative may be calculated by     d d = A(t)(p) A(to + τ )(p) dt dτ t=to τ =0

(∗)

with B1 (τ ) = A(to + τ ) ◦ A(to )−1 , B2 (τ ) = A(to )−1 ◦ A(to + τ ) ⎧   ⎨ A(to + τ ) ◦ A(to )−1 Ato (p) = B1 (τ ) ◦ A(to )(p) A(to + τ )(p) =   ⎩ A(to ) ◦ A(to )−1 ◦ A(to + τ )(p) = A(to ) ◦ B2 (τ )(p) The differentiable mappings τ → B1 (τ ), τ → B2 (τ ) verify B1 (0) = B2 (0) = e. Then (∗) and Proposition 1.11 give i) with     d d = V (q), = W (p). B1 (τ )(q) B2 (τ )(q) dτ dτ τ =0 τ =0 and with q = A(to )(p). The proof of ii) follows the same way. C OROLLARY 1.2.– Let t → A1 (t) and t → A2 (t) be two differentiable maps, A(t) = A1 (t) ◦ A2 (t), let to fixed. Then the elements V and W of [1.89] (for t → A(t) at t = to ) are (with Ai (to ) = Ai , A(to ) = A): V = V1 + A1 ∗ .V2 ,

W = A−1 .W1 + W2 2∗

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55

This property is used for the calculations of the velocity of a compounded motion. 2 For all p ∈ E: 

 d = V1 ◦ A1 (to ) ◦ A2 (to )(p) + A1 (to ) ◦ V2 ◦ A2 (to )(p) A(to + τ )(p) dt τ =0   = V1 + A1 ◦ V2 ◦ A−1 ◦ A1 ◦ A2 (p) 1   = V1 + A1 ∗ V2 ◦ A(p)

C OROLLARY 1.3.– Let t → A(t) be a differentiable map, V and W the elements of [1.89] (for t → A(t) at t = to ). Then the elements V  and W  of [1.89] for the map t → A(t)−1 at t = to are V  = −A(t)−1 ∗ .V,

W  = −A(t)∗ .W

2 Putting A1 (t) = A(t)−1 , A2 (t) = A(t) in Corollary 1.13-1 we derive: V  + A(t)−1 ∗ .V = 0,

 A(t)−1 ∗ .W + W = 0.

In Propositions 1.11 and 1.13 the mappings t → At are curves defining tangent vectors of the manifold D(E) and the results open the way to the study of the group D(E) as a Lie group (as this was done in section 1.5.1 for the matrix form G of D(E)). To the isomorphy of the group D(E) and the matrix group G according to Theorem 1.8 ii) correspond relations between their Lie algebras and their adjoint representations. First of all, assuming the choice of an origin o in E, we shall associate to every skew-symmetric vector field X on E a matrix: M(X) =

∼  ω X X(o) 0 0

so that the value of the vector field X at p may be calculated according to the following rule: 

p ∈ E,

→ x=− op,

  ∼     X x ωv X = X(p) ⇐⇒ = . 0 1 0 0

with v = X(o), ω = ω X . This rule will be justified by the following property (notation of Proposition 1.11): 

 d = M(X), M(At ) dt t=0

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Multi-Body Kinematics and Dynamics with Lie Groups

2 To prove it we take the associated derivative of the time dependent matrix   ∼  −−−−→ d At Ut ω X X(o) with Ut = oAt (o). = 0 0 dt 0 1 t=0 Therefore there is a bijective relation X → M(X) between the set D of skewsymmetric vector fields on E and the set g of section 1.5.1. This relation is a Lie algebra isomorphism and the adjoint representation g → Ad g defined in section 1.5.1.1 for the matrix group G is equivalent to representation A → A∗ (these results are the matter of exercise 1.21 below). E XERCISE 1.20.– Prove Theorem 1.11. H INT.– To prove a) ⇒ b) put, for a fixed p in E, u(x) = X(p + x) − X(p) and use exercise 1.1 of section 1.2. E XERCISE 1.21.– 1) Check that the vectorfield [X, Y ] defined in [1.81] belongs to D, with ω [X,Y ] = ω X ∧ ω Y , and that, endowed with this bracket, D is a Lie algebra (in particular check Jacobi identity). Check formula [1.82]. 2) Prove that relation X → M(X) is a Lie algebra isomorphism between D and g (endowed with the structure defined by the commutator of matrices as in section 1.5.11.5.1 formula [1.58]).In particular check formula:   M [X, Y ] = M(X).M(Y ) − M(Y ).M(X). 3) Prove that if X ∈ D, A ∈ D(E), then Y = A∗ X defined in [1.79] belongs to D and check formulas [1.80] and   M A∗ X = M(A)M(X)M(A)−1 and that the linear representations A → A∗ of D(E) and g → Ad g of G are equivalent. E XERCISE 1.22.– With the notation of Proposition 1.13, put gt = M(At ) for t ∈ I, gto = g = M(A),   d and v = (g, ξ) ∈ Tg G ξ = gt dt t=to (In differential geometry a tangent vector v of the manifold G may be defined as the velocity at time to of a motion described by t → At .) Prove that, with the notation of section 1.5.1.2: ϑr (v) = M(V ),

ϑ (v) = M(W ),

M(V ) = M(A).M(W ).M(A)−1 .

The Displacement Group as a Lie Group

57

E XERCISE 1.23.– Prove that A∗ = identity ⇐⇒ A = e (in other words, for the group D, Ad A = 1 ⇔ A = e: i.e. adjoint representation is faithful, what is not verified for any Lie group). E XERCISE 1.24.– 1) Prove that, if X : E → E is a vector field on E, the following properties are equivalent: a) There exists a real number λ such that:   → → ∀p, q ∈ E : X(q) − X(p) · − pq = λ − pq 2 b) There exist ω ∈ E and λ ∈ R such that: → → ∀p, q ∈ E : X(q) = X(p) + ω ∧ − pq + λ− pq Prove that ω = ω X and λ = λX are uniquely defined. 2) Prove that if (a) is verified with λX = 0 there exists a uniquely defined point γX ∈ E such that X(γX ) = 0. 3) Prove that the set D of the vector fields verifying (a) makes a Lie algebra where the Lie bracket of the vector fields X and Y is defined by [X, Y ](p) = ω X ∧ Y (p) − ω Y ∧ X(p) + λX Y (p) − λY X(p) for all p ∈ E. 4) Check that the Lie algebra D defined in section 1.5.4 with the bracket [1.81] is a Lie subalgebra of D and that [X, Y ] is in D for all X and Y in D. → pq is a skewH INT.– For 1) remark that if (a) is verified, then Xo : p → X(p) − λ− symmetric vector field and use Theorem 1.11. For 2) remark that when λ = 0 the ∼ linear operator λ1+ ω X is invertible. E XERCISE 1.25.– If g is a Lie algebra, a derivation of g is a linear map Δ ∈ L(g) such that       Δ [x, y] = Δ(x), y + x, Δ(y) for all x and y in g. Jacobi identity means that, for u ∈ g the mapping x → ad u.x = [u, x] is a derivation of g (inner derivation defined by u) but, in general, there exist derivations which are not of this form. The target of this exercise is to determine the derivations of the Lie algebra D using the Lie algebra D introduced in exercise 1.24. 1) Check that for all U ∈ D the mapping X → [U, X] (with the bracket defined in D) is a derivation of D.

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Multi-Body Kinematics and Dynamics with Lie Groups

2) Check that all the derivations of the Lie algebra (E, ∧) are inner derivations. 3) Let o ∈ E and splits D as D  E × E by the linear isomorphism X → (ω X , X(o)) so that every linear operator Δ in D is described by a bloc matrix as       ωX AB ωX → × X(o) CD X(o) where A, B, C, D are linear operators in E. Check that the necessary and sufficient conditions for Δ be a derivation is that the following two properties be verified: i) A and C are derivations of the Lie algebra (E, ∧). ii) For all x and y in E: D(x ∧ y) = A(x) ∧ y + x ∧ D(y). 

4) Deduce that i) and " ii) is equivalent to say that the matrix of Δ is of the form  !∼ a 0 AB with a and c in E and λ in R. = ∼∼ CD c a +λ1

5) Check that all the derivations of the Lie algebra D are of the form X → Δ(X) = [U, X] with a uniquely defined U ∈ D. H INT.– For 1), take an orthonormal frame of E. For 4) prove that if L is a linear operator in E such that for all x and y: L(x ∧ y) = x ∧ L(y) then there exist λ ∈ R such that L = λ1. Take L = D − A. In order to expand the differential calculus on D in the form of calculus on Lie groups we first show that D appears as a submanifold of an affine space, this manifold and its tangent vectors being defined as it is explained in section 1.1. This is the matter exercises 1.26 and 1.27 below. E XERCISE 1.26.– Let A(E, E) be the set of affine maps from E to E and A(E, E) be the set of affine maps from E to E. 1) Check that A(E, E) is a vector space over R and that D is a vector subspace of A(E, E). Of course the operations in this vector space are defined by: (X + Y )(p) = X(p) + Y (p),

(λX)(p) = λX(p) for p ∈ E

for X and Y in A(E, E) and λ ∈ R. −−−−−−→ 2) Prove that if A and B are in A(E, E) the map p → A(p)B(p) from E to E is in A(E, E) and its linear part is B − A. 3) Check that A(E, E) is an affine space over the vector space A(E, E) if we define −−−−−−→ −−→ the vector AB as the vector field p → A(p)B(p)

The Displacement Group as a Lie Group

59

E XERCISE 1.27.– The group D is considered as a subset of the affine space A(E, E) of exercise 1.26 1) * Give a formal proof that D = D(E) is a submanifold of the affine space A(E, E) and a Lie group. 2) According to question 1) and section 1.1, a tangent vector of D at A is a pair (A, X) such that X ∈ A(E, E). Show that when A ∈ D, the following properties are equivalent: – (A, X) ∈ TA D, – there exists V ∈ D such that X = V ◦ A, – there exists W ∈ D such that X = A ◦ W , and show that under these conditions V = A∗ W . 3) Check that Te D is isomorphic to the vector space {e} × D = {(e, X) | X ∈ D} and, finally to D. H INT.– For question 1) prove that the map A → A∗ ◦ A (where A is the linear part of A), is a submersion from a suitable open subset of A(E, E) on to the a suitable open subset of the vector space Ls (E). For question 2) consider a differentiable curve t → At in D such that Ao = A, defining a tangent vector v ∈ TA D, and use Proposition 1.13. E XERCISE 1.28.– Let G ∈ D and G be its linear part. The left and right translations by G in D are the maps from D onto D such that LG : A → G ◦ A, RG : A → A ◦ G. 1) Check that, these maps are differentiable and that, if v = (A, X) ∈ TA D is the T tangent vector to the curve t → At with A0 = A, then the tangent maps LTG and RG verify LTG (v) = (G ◦ A, G ◦ X),

T RG (v) = (A ◦ G, X ◦ G)

and that, in the conditions of exercise 1.27 question 2 and Proposition 1.13, the general process to define the Maurer-Cartan form on D reads: ϑ (v) = LTA−1 (v) ≡ (e, W ),

T ϑ (v) = RA −1 (v) ≡ (e, V ) for v ∈ TA D

when  v = (A, X) ≡

(A, V ◦ A) with V ∈ D (A, A ◦ W ) with W ∈ D

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Multi-Body Kinematics and Dynamics with Lie Groups

so that, if the Lie group D is considered as a submanifold of A(E, E) and {e} × D is identified with D, the Maurer-Cartan forms of D are given by ϑ (v) = A−1 X ≡ W,

ϑr (v) = XA−1 = V

[1.91]

2) Check that, if the motion of a rigid body and its velocity at time t = 0 are defined by the map t → At with Ao = A and the tangent vector v ∈ TA D: 

d v= At dt

 (where t=to

d understands as “tangent map”) dt

then, according to the definition of the Maurer-Cartan forms, the elements V and W of Theorem 1.13 are ϑr (v) = V , ϑ (v) = W. 3) Check that the adjoint representation of D in D, defined according to the general T theory of Lie groups by Ad G.u = LTG−1 ◦ RG −1 (u) for u ∈ Te D, is expressed by: Ad G.X = G∗ X if u = (e, X) with X ∈ D is identified with X. and that: ϑr (v) = Ad A.ϑ (v) for all v ∈ TA D 1.5.5. The Klein form on D The inner product on D corresponding to the inner product defined in section 1.5.1.5 is expressed by [[X | Y ]] = ω X · Y (a) + ω Y · X(a) for X, Y ∈ D,

[1.92]

where the choice of a ∈ E is immaterial. Then [[· | · ]] is a non-degenerate bilinear form on D and, for X, Y , Z ∈ D and A ∈ D: [[[Z, X] | Y ]] + [[X | [Z, Y ]]] = 0,

[1.93]

[[A∗ X | A∗ Y ]] = [[X | Y ]],

[1.94]

(what is corresponding to formulae [1.72] and [1.73] by isomorphy). Formula [1.93] also means that the map (X, Y, Z) → [[X | [Y, Z]]] is a skew-symmetric trilinear form on D.

[1.95]

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61

P ROPOSITION 1.12.– Let X and Y ∈ D − T. Assume that one of these assumptions is verified a) ΔX and ΔY are orthogonal (i.e. ω X · ω Y = 0), b) ΔX and ΔY are not parallel (i.e. ω X ∧ ω Y = 0) and [[X | X]] = [[Y | Y ]] = 0, Then [[X | Y ]] = 0 ⇐⇒ ΔX and ΔY lie in the same plane. In screw theory, orthogonality for the Klein form is called “reciprocity of screws” (see [BAL 00]). But with no special assumptions about X and Y a simple geometrical interpretation of relation [[X | Y ]] = 0 would not exist. The proof of the proposition is the matter of a part of exercise 1.31. E XERCISE 1.29.– Check formula [1.93] and [1.94] by direct calculations. Prove that indeed [1.93] may be derived from [1.94] and that Jacobi identity in D may be derived from properties of the representation A → A∗ . H INT.– see the method used in the proof of Proposition 1.11. E XERCISE 1.30.– 1) Check that the Klein form is a non-degenerate bilinear form verifying [1.93] and [1.94]. 2) Prove that the isotropic vectors of D for the Klein form are the vectors of Z ∪ T with # Z= Za . [1.96] a∈E

E XERCISE 1.31.– Let X and Y be in D − T and let δ be the common perpendicular to their axes ΔX and ΔY (or any common perpendicular when they are parallel), ϕ the − → angle between ΔX and ΔY . Let a = δ ∩ ΔX , b = δ ∩ ΔY and d = ab . Check that:  − →  [[X | Y ]] = (fX + fY )(X | Y ) − ab · ω X ∧ ω Y ,    = fX + fY cos ϕ − d sin ϕ ω X ω X ,

[1.97]

(formula [1.97] is a classical formula of “screw-theory”) and prove Proposition 1.12. E XERCISE 1.32.– Let (o; e1 , e2 , e3 ) be an othonormal right-handed frame of E. Define six elements of D by 

ξ i (o) = 0, ω i = ei for i = 1, 2, 3, ξ i+3 (0) = ei , ω i+3 = 0 for i = 1, 2, 3.

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1) Prove that the ξ i for i = 1, . . . , 6 are isotropic for the Klein form, make a basis of D and that the  6 ×6 matrix of the bilinear Klein form with respect to this basis is 01 the bloc matrix where 0 and 1 are the null and unit matrices 3 × 3. 10 2) Define six elements of D by ⎧ 1 ⎪ ⎪ ⎪ $i = √2 (ξ i + ξi+3 ) ⎨

for i = 1, 2, 3,

⎪ ⎪ 1 ⎪ ⎩ $i+3 = √ (ξ i − ξi+3 ) for i = 1, 2, 3. 2 Prove that ($i | i = 1, . . . , 6) is an orthogonal basis of D for the Klein form and check that the signature of the Klein form is (3+, 3−). The following query is linked with existence of an objective measure of “torsors” used in applied mechanics, (and it is of practical importance, see [DUF 96] in the preface or [DUF 90]). Is there a symmetric non-degenerate positive definite form defined on D and which is invariant by adjoint action (that is to say an Euclidean structure on D verifying [1.94])? (It may be proved that the answer is no, see Chap 2, section 2.2.2, exercise 2.10.). 1.5.6. Operator  D EFINITION 1.6.– A linear map  : D → D is defined by: X = the constant vector field equal to ω X on E. Operator  will be used, in an equivalent form, in the chapter developing the applications of dual numbers to kinematics. The range and the kernel of  are both equal to T, hence it verifies 2 = 0. Moreover, for all X, Y ∈ D and A ∈ D: [X, Y ] = [X, Y ] = [X, Y ],

[[X | Y ]] = [[X | Y ]],

A∗ (X) = A∗ X. [1.98]

2 According to the definitions of the Lie bracket,  and the Klein form [X, Y ], [X, Y ], [X, Y ] are all equal to the constant vector field equal to ω [X,Y ] = ω X ∧ ω Y on E, [[X | Y ]] and [[X | Y ]] are both equal to ω X · ω Y . According to [1.79] and [1.80], since X is the constant  field  equal to ω X , A∗ (X) is the constant field equal to A(ω X ), since ω A∗ X = A ω X , A∗ X is the same constant field. The following theorem is an algebraic form of Theorem 1.12:

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T HEOREM 1.14.– Let X ∈ D with X ∈ / T. There are uniquely defined Z, U such that (see [1.96]): X = Z + U,

Z ∈ Z,

U ∈ T,

[X, U ] = 0.

Moreover, U = fX X with fX defined in Theorem 1.12. R EMARK 1.5.– If X = Z + U the following properties are equivalent: [X, U ] = 0,

[Z, U ] = 0,

[Z, X] = 0.

The third condition means that X and Z have the same axis, the first two conditions means that the direction of U is the direction of the axes of X and Z. 2 First of all note that if U ∈ T, X ∈ / T: [X, U ] = 0 ⇐⇒ U = λX with λ ∈ R. Then, according to the result of exercise 1.30, if X = Z + U : Z ∈ Z ⇐⇒ [[X − λX | X − λX]] = 0 ⇐⇒ [[X | X]] − 2λω 2X = 0 ⇐⇒ λ =

1 [[X | X]] . 2 ω 2X

Hence U necessarily takes the value mentioned in the Theorem and it is uniquely defined as Z. E XERCISE 1.33.– – Prove that when X ∈ / T: Ker ad X = {αX + βX | α, β ∈ R}, Im ad X = {Z ∈ D | ω Z · ω X = 0, and [[X | Z]] = 0}, D = Ker ad X ⊕ Im ad X. – Prove that when X ∈ T: Ker ad X = {Z ∈ D | Z is colinear with X}, Im ad X = {Z ∈ D | [[X | Z]] = 0},

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E XERCISE 1.34.– Let us define the Killing form of D as the bilinear form (X, Y ) → (X | Y ) = [[X, Y ]] (= ω X · ω Y ). 1) Check that, for X, Y , Z, T ∈ D [X, [Y, Z]] = [[X | Z]] Y − [[X | Y ]] Z + (X | Z)Y − (X | Y )Z.

[1.99]

[[ [X, Y ] | [Z, T ]]] = (X | Z)[[Y | T ]] + (Y | T )[[X | Z]] −(X | T )[[Y | Z]] − (Y | Z)[[X | T ]] (formula [1.99] is an important generalization to D of Gibbs formula of ordinary vector algebra. It will take a remarkable form [2.4] section 2.1.4 in the dual number setting).   2) Prove that Tr ad X ◦ ad Y = −2(X | Y ) where Tr means trace of linear operator (the left-hand side is the general definition of the Killing form of a Lie algebra; the definition at the head of this exercise is used in “screw theory”). H INT.– Use exercise 1.2. E XERCISE 1.35 (Axis of A∗ X.).– Let X ∈ D such that ω X = 0 and A ∈ D, Y = A∗ X. Let ΔX (directed as ω  X ) and ΔY (directed as ω Y ) be their axes. Prove that as directed lines ΔY = A ΔX . E XERCISE 1.36 (Axis of the Lie bracket.).– The notation is that of exercise 1.31. 1) Prove that, when ΔX and ΔY are not parallel, the axis of [X, Y ] is δ (oriented by a normed vector k such that (ω X , ω Y , k ) is a right-handed basis). 2) Check that, for p ∈ δ: − → [X, Y ](p) = (fX + fY )ω X ∧ ω Y + (X | Y )ab.

[1.100]

− → If d = ab is the distance between the axes and ϕ is their angle the formula reads:   [X, Y ](p) = (fX + fY ) sin ϕ + d cos ϕ ω X ω Y k. Check f[X,Y ] (when the axes are not parallel) and the value of [X, Y ] when the axes are parallel. 3) Prove Proposition 1.10. H INT.– For 1) use Proposition 1.12.

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E XERCISE 1.37 (Axes of linear combinations.).– The notation is that of Exercise 1.31. Let X and Y ∈ D − T such that [X, Y ] ∈ / T (i.e. ΔX and ΔY are not parallel). 1) Prove that for all real numbers λ and μ (not both equal zero) the axis of λX +μY meets δ. 2) Let S be a two-dimension vector subspace of D and Ω = {ω X | X ∈ S}. We assume that Ω is a two-dimension vector subspace of E. Prove (with no calculation) that there is a basis (ξ, η) of S such that (ξ | η) = 0 and [[ξ | η]] = 0 3) We assume that ω X = ω Y = 1 and that k is the normalized vector in the direction of ω X ∧ ω Y so that ω X ∧ ω Y = sin ϕ k. Let c be the intersection of δ with the axis of λX + μY . Check that: i) If a = b then c = a + rk with: r=

1 λμ(fY − fX ) sin ϕ. G

ii) If a = b, then c is the barycenter c = sa + tb (s + t = 1) such that, putting G = λ2 + μ2 + 2λμ cos ϕ:  1  λμ μ(μ + λ cos ϕ) − (f − fX ) sin ϕ , G d Y  1  λμ t= λ(λ + μ cos ϕ) + (f − fX ) sin ϕ , G d Y μ(μ + λ cos ϕ) λ(λ + μ cos ϕ) λμ c= a+ b+ (f − fX ) sin ϕ k. G G G Y

s=

4) We assume that ω X = ω Y = 1, ω X · ω Y = 0 and that k is the normalized vector in the direction of ω X ∧ ω Y . For α ∈ R let ξ = cos αX + sin αY,

η = − sin αX + cos αY.

→ → Let o be the middle of ab, express − oc ξ and − oc η and give another proof of the existence of ξ and η verifying the conditions of question 2).

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5) Let S be the ruled surface generated by the axes of the elements of S (axode of S). Prove that, taking the axes in the natural manner, this surface is defined by the parametric equations x = ρ cos α,

y = ρ sin α,

ρ ∈ [0, +∞[,

α ∈ [0, 2π].

z=

d f − fY cos 2α + X sin 2α 2 2

or by the Cartesian equation z=

xy d x2 − y 2 + (fX − fY ) 2 , 2 x2 + y 2 x + y2

(“Plücker’s Cylindroïd or “Plücker’s Conoïd”). H INT.– For 1) use Proposition 1.12 and [1.93]. For 2) remark that (X, Y ) → (X | Y ) = ω X · ω Y defines an Euclidean structure on Ω and apply a standard result on symmetric bilinear forms in Euclidean vector spaces to the map induced on S by the Klein form. For 4) prove that  1 − → → oc = oc η d cos 2α + (fX − fY ) sin 2α = −− ξ 2 E XERCISE 1.38 (Morley-Petersen Theorem).– Let Δ1 , Δ2 , Δ3 be lines such that both of them are not parallel. Let δ1 , δ2 , δ3 be the respective common perpendicular to Δ2 and Δ3 , to Δ3 and Δ1 , to Δ1 and Δ2 . Let D1 , D2 , D3 be the common perpendicular to Δ1 and δ1 , to Δ2 and δ2 , Δ3 and δ3 . Prove that D1 , D2 , D3 have a common perpendicular (Morley-Petersen Theorem). H INT.– Use exercises 1.36 question 1) and 1.37 question 1) and Jacobi identity. 1.5.7. One-parameter subgroups of D and exponential mapping D EFINITION 1.7.– A one-parameter subgroup of D is a differentiable map G : R → D such that: ∀s and t ∈ R : Gt+s = Gt ◦ Gs Then G is an homomorphism of the additive group (R, +) into D, the range of G is a commutative subgroup (also named one-parameter group) and, of course, Gt ◦Gs = Gs ◦ Gt , Go = e. There is a one-one relation between one-parameter groups of D and elements of the Lie algebra D:

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T HEOREM 1.15 (Infinitesimal generator of a one-parameter subgroup).– a) If t → d  Gt is a one-parameter subgroup of D and X = dt Gt t=0 (X ∈ D), then t → Gt is solution of a “Cauchy problem” (differential equation with initial condition): ∀ t:

d Gt = X ◦ Gt , dt

G0 = e

(∗)

b) Conversely, for any X ∈ D, the “Cauchy problem” (∗) has one and only one maximal solution defined on R and this solution GX : t → GX t is a one-parameter subgroup of D. (Remind that the maximal solution of a “regular” differential equation with initial condition at t = 0 is unique but might be defined only on a smaller domain than R.) 2 Existence of X is a consequence of the differentiability and Proposition 1.11. To prove (∗), formula [1.84] at Gt (p) rather than p, leads to:         d d d = = X Gt (p) Gt (p) = Gt+s (p) Gs Gt (p) dt ds ds s=0 s=0 First of all let us prove (b) when X ∈ T and X(p) = u for all p ∈ E. The Cauchy problem (∗) reads ∀t, ∀p ∈ E :

d Gt (p) = u, dt

G0 (p) = p

and it has the only solution Gt (p) = p + tu, that is GX t = Tr (tu). In particular when X = 0, G0t = e for all t. Now take X ∈ D − T, use Theorem 1.12 and choose an origin o on ΔX so that, for all a ∈ E: → oa, X(a) = X(o) + ω X ∧ −

X(o) = fX ω X

−−−−→ Put oGt (p) − tX(o) = ξ t where t → ξ t (∈ E) is the new unknown map. Then ⎧ ⎧   d d ⎪ ⎪ ⎨ ξt = ωX ∧ ξt ⎨ Gt (p) = X Gt (p) dt dt ⇐⇒ ⎪ ⎪ ⎩ ⎩ → G0 (p) = p ξ0 = − op ∼ → → op = R(u, t ω X ).− op, where R(u, θ) is the vectorial Therefore ξ t = exp tω X . − rotation in E about u = ω/ ω , a normalized director of ΔX (see Theorem 1.2). All thing considered when o ∈ ΔX :

 → →  −−−−→ op = R(u, t ω ). − op + tX(o) oGt (p) = tX(o) + R u, t ω .−

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(since ω X ∧ X(o) = 0), what is equivalent to     GX t = Tr (tfX ω X ) ◦ Rot ΔX , t ω X = Rot ΔX , t ω X ◦ Tr (tX(o))

[1.101]

This formula defines a one-parameter subgroup the element of which are helical motions about the axis of X, the pitch (translation along the axis after one turn) of which is 2πfX . It is easy to check that GtX = GX st for s, t ∈ R (with unicity of the maximal s X solution to a Cauchy problem) so that GtX 1 = Gt . This remark leads to the following definition: D EFINITION 1.8.– The exponential map of the group D is the map expD : D → D (denoted by exp when there is no ambiguity) such that expD X = GX 1 . Then the oneparameter group generated by X ∈ D is expressed as t → expD tX and, by definition: d expD (tX) = X(expD (tX)). dt With the notation of general of Lie group theory, the definition of the exponential map ϑr

d  expD (tX) = X for all t dt

[1.102]

It is not true that expD X = expD Y implies X = Y (see [1.101]) but, at the level of relation between one-parameter groups and infinitesimal generator, Theorem 1.15 leadsto: ∀t ∈ R : expD tX = expD tY ⇐⇒ X = Y Regarding the operations defined in D, for A ∈ D, X ∈ D, we have the following properties  −1 expD (−X) = expD X , expD 0 = e   expD A∗ X = IntA. expD X

[1.104]

(expD X)∗ = exp ad X

[1.105]

d expD (tX) Y = [X, expD (tX) Y ] = expD (tX) [X, Y ] dt

[1.106]

[1.103]

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Formula [1.106] also reads: d Ad expD tX = ad X ◦ expD tX = expD tX ◦ ad X for all t dt

[1.107]

Formula [1.105] is an equality in the linear group Gl(D), to the right ad X is the linear operator Y → [X, Y ] in D and exp is the ordinary exponential of operators. 2 Properties [1.103] are evident. For proving [1.104] define the map t → Ft = IntA. expD (tX) (∈ D) then, for all p ∈ E, F0 (p) = p and: d Ft (p) = A ◦ X ◦ expD (tX) ◦ A−1 (p) = Y (Ft (p)) with Y = A∗ X dt Therefore the maps t → GYt (p) and t → IntA.GX t (p) are solutions of the same differential equation with the same initial value at t = 0. Since the maximal solution of the Cauchy  problem is unique [1.104] follows. To prove [1.105] consider the map t → Φt = expD tX)∗ from R, to L(D), then (see [1.90]): d Φt = ad X ◦ Φt , dt

Φ0 = identity

The map t → Φt is a solution of a Cauchy problem in L(D) which has one and only one solution t → exp(t ad X) where exp means the exponential in L(D) and the result follows. Now from [1.105], exp(tX)∗ X = X for all t and [1.106] follows from [1.82]. There is no simple general expression of GX+Y , exp t(X + Y ) and exp(tX) ◦ exp(tY ) for any X and Y and D. P ROPOSITION 1.13.– Let X and Y ∈ D, then the following properties are equivalent a) For all s and t: expD (tX) ◦ expD (sY ) = expD (sY ) ◦ expD (tX), b) [X, Y ] = 0.  X −1 Y = GYs or 2 Relation (a) reads for all s and t: GX t ◦ Gs ◦ Gt   X Y Y ∀ s, t : Int GX t .Gs = Gs ⇐⇒ ∀ s, t : expD (Gt )∗ sY = expD (sY ) ⇐⇒ ∀ t : (GX t )∗ Y = Y =⇒ [X, Y ] = 0 (taking the derivative for t = 0). Hence (a) ⇒ (b). To prove the converse it is sufficient to show that the derivative of the map t → exp(tX) ◦ exp(sY ) ◦ exp(−tX) vanishes. Therefore this map is equal to its value for t = 0.

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The exponential mapping is a tool for expressing the members of a neighborhood of identity in D and in all the subgroups of the displacement group as products of exponentials. Rotary joints, prismatic joints and helical joints in mechanical systems are described by one-parameter groups and in [HER 99] pp. 212-233, J. Hervé presents the various manners to synthesize mechanisms moving according to any subgroup of D with those joints. However, except in specific situations (as investigations on singularities of mechanisms), the representation of the displacements by exponentials of one element of D is not really efficient for calculations. A reason is that, expressed with the infinitesimal generators, the law of the group is given by Campbell-Hausdorff-Dynkin formula exp X exp Y = exp H(X, Y ) where H(X, Y ) is an exceedingly complicated expression (see. [CHE 06], Chap IV). E XERCISE 1.39.– 1) Deduce [1.104] and [1.105] from formula [1.101]. 2) Deduce (b) ⇒ (a) in Proposition 1.13 from a direct calculation of Int expD (tX) . expD sY .   3)* Show that [X, Y ] = 0 =⇒ expD X ◦ expD Y = expD Y ◦ expD X but that the converse is untrue. E XERCISE 1.40.– 1) Check that when U ∈ T is the constant field equal to u ∈ E, then expD U = Tr (u) ,

exp ad U = 1 + ad U = (expD U )∗

2) Check that when X ∈ Za , expD is a rotation about a in E. E XERCISE 1.41 (Euler’s angles (affine version)).– This exercise completes exercise 1.10, Chap 1. Let (o; i, j, k) be an orthonormal affine frame of E and let ξ, η, ζ be the normalized elements of zo such that, for example ξ(o) = 0, ω = i. Let ψ, ϑ, ϕ be ξ real variables (angles), put: A = expD (ψζ),

ξ o = Ad A.ξ,

η o = Ad A.η,

B = expD (ϑξ o ),

Z = Ad B.ζ,

C = expD (ϕZ),

X = Ad C.ξ 0 ,

Y = Ad CB.η o ,

G(ψ, ϑ, ϕ) = C ◦ B ◦ A (∈ Rot(o)).

1) Check that Y = [Z, X] and that (X, Y, Z) is a basis of zo . 2) Using the properties of adjoint representation and exponential map of D, prove that G(ψ, ϑ, ϕ) = expD ϕζ ◦ expD ϑξ ◦ expD ψζ. 3) Prove that the map (x, y, z, ψ, ϑ, ϕ) → Tr (xi + yj + zk) ◦ G(ψ, ϑ, ϕ) defines a chart of the manifold D on an appropriate neighborhood of e.

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1.6. Conclusion The calculations of kinematics and dynamics of systems of rigid bodies operate on a Lie group G and its Lie algebra g, in practice the Euclidean displacement group D (or a certain subgroups as rotation groups). They refer to mathematical structures ordered in three principal levels: Level 1 refers only to the structure of Lie group (and its Lie algebra) of G in coordinate-free form. Level 2 uses particular features of D or its Lie algebra D (also in coordinate-free form). Level 3 uses coordinates and matrix representations. Above, through examples we have pointed out the properties belonging to levels 1 and 2. Of course all of them would appear at level 3 in an expanded form (when they are not unfortunately hidden!). The most compact form of computations lies at level 1 and therefore it is worthwhile to perform most of them at level 1 and after to translate them to levels 2 or 3 if (and only if!) this appears to be necessary. However, a lot of classical methods go into mechanics of multibody systems directly at level 3 and try to bring out interesting properties; in our opinion this not the best starting point for a deep insight into their mathematical origin. Let us sum up the properties of Level 1 for a general Lie group G and its Lie algebra g (what would be useful to mechanics with G = D and its Lie algebra g = D). 1) G has a structure of manifold allowing differential calculus ; the operations of the group are differentiable. Moreover, for h ∈ G, the left and right translations Lh : g → h.g and Rh : g → g.h are diffeomorphisms so that their tangent maps define vector space isomorphisms between the tangent spaces Te G and Th G. 2) The tangent vector space at identity Te G may be endowed with a Lie algebra structure denoted by g. Indeed, as a set, g may be exactly Te G or another set the choice of which is the most convenient (see in [CHE 06] the various ways to define the Lie algebra in general theory). Let [·, ·] be the Lie bracket, we have:  (u, v) → [u, v] is a skew-symmetric bilinear map [1.108] [u, [v, w]] + [v, [w, u]] + [w, [u, v]] = 0, (Jacobi identity) 3) The group G acts on its Lie algebra g by the adjoint representation Ad : ⎧ Ad g is a linear operator in g ⎪ ⎪ ⎨ Ad e = identity [1.109] ⎪ Ad g1 g2 = Ad g1 ◦ Ad g2 ⎪ ⎩ Ad g.[u, v] = [Ad g.u, Ad g.v]

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4) Two g-valued differential forms ϑr and ϑ , the Maurer-Cartan forms, define linear isomorphisms from Tg G onto g  Te G for every g and they verify: ϑr (LTh v) = Ad h.ϑr (v), ϑr (RhT v) = ϑr (v) ϑ (LTh v) = ϑ (v), ϑ (RhT v) = Ad h−1 .ϑ (v)

[1.110]

The differential of Ad g at v ∈ Tg G is expressed by: Dg (Ad g.x)(v) = ad ϑr (v).Ad g.x = Ad g.ad ϑ (v).x

[1.111]

The exterior differential these forms are expressed by Maurer-Cartan formulae: dϑr = [ϑr , ϑr ],

dϑ = −[ϑ , ϑ ]

[1.112]

5) There is an exponential mapping exp : g → G generating “finite elements of G“ from “infinitesimal” elements. In general this map is neither one-one neither onto but there always exist an open neighborhood U of 0 in g and an open neighborhood O of e in G such that exp induces a diffeomorphism U → O. Moreover: Ad (exp u) = exp(ad u) exp(Ad g.u) = g.(exp u).g −1

[1.113]

For the classical groups the exponential mapping may be expressed in closed form; Olinde-Rodrigues formula for SO(E) or the calculations in section 1.5.1.6 for D(E) are examples. And yet it is a tool for theoretical investigations of the structure of Lie groups of little use to kinematics except in specific situations (see section 1.5.7). At level 2 we find properties of the Euclidean group D as the properties of “torsors” linked with the algebraic properties of D: 6) The group D is a semi-direct product of the rotation group about any point and the translation group. Its Lie algebra splits into D = T ⊕ Za and it is endowed with the Klein form [[· | ·]], which is an inner product (symmetric non-degenerate bilinear form) verifying: [[Ad g.u | Ad g.v]] = [[u | v]]

[1.114]

(u, v, w) → [[u | [v, w]]] is a skew-symmetric multilinear form of degree 3 on D.

[1.115]

The Klein form is not positive, its signature is (+ + + − −−), in classical kinematics orthogonality for this form is called “reciprocity”, in dynamics it appears in the definition of work.

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An operator  with the properties which were described in section 1.5.6 is defined; these properties open the way to the dual number methods in kinematics (see Chapters 2 and 3). 1.7. Appendix 1: The algebra of quaternions 1.7.1. First definition of quaternions E denotes the oriented Euclidean vector space in dimension 3. The first form of definition consists in endowing the vector space H = R × E with a product defined by: qq = (ss − v · v , sv + s v + v ∧ v ) si q = (s, v), q = (s , v ). Endowed with its natural vector space structure and this product H is a R-algebra with unity e = (1, 0) ; in fact it is easily verified that q(q q ) q(q + q ) (q + q )q λ(qq ) eq

= (qq )q , = qq + qq , = (q + q )q, = (λq)q = q(λq ), = qe = q.

In particular, the mapping defined by this product (q, q ) → qq is R-bilinear. It will appear in the following that, endowed with addition and this product H is a non-commutative field. Two natural projections are defined on H: H → R : q = (s, v) → s = Re (q), H → E : q = (s, v) → v = Ve (q), The number Re (q) is the real part of quaternion q and Ve (q) its vectorial part. A quaternion is said to be real when its vectorial part vanishes and purely vectorial or pure when its real part vanishes. Two natural injective mappings are defined: R → H : s → (s, 0) j : E → H : x → (0, x) = [x]. Their respective ranges [R] and [E], are made of the real and pure quaternions and the may be “identified” with R and E. In particular we shall often note 1 the unit element e = (1, 0). We have: H = [R] ⊕ [E].

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For λ, λ ∈ R, q, q ∈ H, we have: Re (qq ) = Re (q q), Re (λq + λ q ) = λRe (q) + λ Re (q ), Ve (λq + λ q ) = λVe (q) + λ Ve(q ). 1.7.2. Center of H The center of H is made of the quaternions which are commuting with every other quaternion. According to the definition of the product od quaternions, in order that qq = q q, it is necessary and sufficient that Ve (q) and Ve (q ) be linearly dependent vectors. Therefore, in order that q commute with every quaternion, it is necessary and sufficient that Ve (q) = 0: the center of H is equal to [R]. 1.7.3. Conjugation in H A natural involution is defined in H by ¯ = (s, −v), (conjugate of q) q = (s, v) → q and it is an antiautomorphism of the algebra H, namely a R- linear mapping verifying, for q1 , q2 , q = (s, v) ∈ H: ¯1 ¯1 + q ¯ 2 , q1 q 2 = q ¯2q q1 + q 2 = q ¯ = (2s, 0) = 2 Re (q)(real quaternion), q+q ¯ = (0, 2v) = 2 Ve (q)(pure quaternion), q−q ¯ q = (s2 + v2 ) e(real quaternion). q¯ q=q 1.7.4. Euclidean structure of H ¯ 2 ) = Re (q2 q ¯ 1 ) = s1 s2 +v1 ·v2 . Therefore, When q1 et q2 ∈ H we have: Re (q1 q as a vector space over R, H is endowed with an Euclidean structure, the scalar product of which is the positive definite symmetric bilinear form: ¯ 2 ). (q1 | q2 ) = Re (q1 q The associate norm is (identifying [R] and R): |q| =

√

q¯ q=

√

¯q = q



s2 + v 2 .

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75

Moreover, this scalar product and the associate norm verify: ¯ ), (aq1 | q2 ) = (q1 | a ¯ q2 ) (q1 a | q2 ) = (q1 | q2 a |q1 q2 | = |q2 q1 | = |q1 | |q2 |, |¯ q| = |q|. The first two formulas mean that the adjoint mappings of the R-linear mappings δa : q → qa and γa : q → qa, are: δa = δa¯ , γa = γa¯ . Those properties have important consequences. First, every non zero quaternion q has an inverse q−1 , such that qq−1 = q−1 q = e, defined by: q−1 =

1 ¯, q |q|2

what is proving that H is a field. Moreover |q−1 | = |q|−1 Second H − {0} = H∗ is a group for the multiplication of quaternions (property of the complement of 0 in any field). More particularly, let U = {q ∈ H | | q |= 1} be the set of normalized quaternions. Then U is a group playing an important role in the representation of rotations in E (when q1 and q2 ∈ U there is |q1 q−1 2 | = |q1 | |q2 |−1 = 1). Let us also note several formulas regarding pure quaternions: [x] = −[x],

([x] | q) = x · Ve (q),

([x] | [y]) = x · y.

meaning that the adjoints of the R-linear mappings j and Ve are: j  = Ve , (Ve) = j. And also: Re ([x][y])

= −x · y,

Re ([x][x ][x ]) = −(x, x , x ), ([x] | [y])

= x · y,

Ve ([x][y])

= x ∧ y,

Ve ([x][x ][x ]) = x ∧ (x ∧ x ), [1.116] [x]

= − [x] .

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1.7.5. Second definition of quaternions Let us consider the vector space H = R4 and its canonical basis (e0 , e1 , e2 , e3 ). We may define a bilinear multiplication H × H → H, by the following table: e0 e0 e1 e2 e2 e3 e3 e1

= = = =

e0 e3 e1 e2

, = = =

e1 e1 = e2 e2 = e3 e3 = −e0 , −e2 e1 , −e3 e2 , −e1 e3 .

If q = q0 e0 + q1 e1 + q2 e2 + q3 e3 , q = q0 e0 + q1 e1 + q2 e2 + q3 e3 , then qq is defined by: qq = (q0 q0 − q1 q1 − q2 q2 − q3 q3 ) e0 +(q0 q1 + q1 q0 + q2 q3 − q3 q2 ) e1 +(q0 q2 + q2 q0 + q3 q1 − q1 q3 ) e2 +(q0 q3 + q3 q0 + q1 q2 − q2 q1 ) e3 It would be easy to verify directly that this multiplication is associative, that, endowed with its vector space structure over R and this multiplication, H is a Ralgebra and that, endowed with addition and multiplication, it is a non-commutative field the unity of which is e = e0 . The real and vectorial parts of the quaternion q and its conjugate may be defined as: Re (q) = q0 ∈ R,

Ve (q) = q1 e1 + q2 e2 + q3 e3 ∈ R3 ,

¯ = q0 e0 − q1 e1 − q2 e2 − q3 e3 ∈ R4 . q If we take up the first form of definition of quaternions, with H = R × E, and if we choose a right-handed orthonormal basis B = (e1 , e2 , e3 ) in E, as it is easily seen relation: q = (s, v) ∈ R × E → (s, v1 , v2 , v3 ) ∈ R4 is an isomorphism for the structures of R-algebras or for the structures of fields. This isomorphism agrees with operations as taking the real or vectorial part or conjugation. Therefore the second form of the definition expresses the first form when a righthanded orthonormal basis is chosen in E and the real and vectorial parts are expressed with their coordinates. Let us note that the definition which is always understood in this work is the first form, based on R×E = H and assuming no choice of coordinates in E.

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1.8. Appendix 2: Lie subalgebras and ideals of D Let E be the three-dimension Euclidean oriented vector space endowed with the vector product and (E, ∧) the corresponding Lie algebra. To derive the classification of the Lie subalgebras of D = D(E) we shall refer to the following lemmas, which may be proved as exercises: L EMMA 1.6.– The Lie subalgebras of (E, ∧) are the subspaces of the form {0},

Ru with u = 0 ,

E

The ideals of (E, ∧) are {0} and E. The second property means that the Lie algebra (E, ∧) is simple. L EMMA 1.7.– Let X ∈ D with X ∈ / T. Then the linear subspace of T which are invariant under the action of ad X are the subspaces of the list: {0},

RX,

T ∩ X ⊥,

T

where Xis the constant vector field equal to ω X on E and X ⊥ is the orthogonal subspace of X with respect to the Klein form. E XERCISE 1.42.– 1) Verify that the vector spaces of the list are invariant under the action of ad X. 2) Prove that if V is a subspace of T invariant by ad X which is neither equal to {0} nor equal to RX then it is equal to T ∩ X ⊥ or T. H INT.– 1) Remark that RX = T ∩ Ker ad X and Im ad X ⊂ X ⊥ . 2) Remark that there exists Z = 0 in V such that Z and ad X.Z ∈ T ∩ X ⊥ . Deduce that T ∩ X ⊥ is a linear subspace of T of dimension 2 contained in V. T HEOREM 1.16 (Classification of subalgebras and ideals of D).– i) The Lie subalgebras of D are of the forms: – D, – Za (= {X ∈ D | X(a) = 0}) with a ∈ E,   / T, – RX, RX ⊕ RX, RX ⊕ T ∩ X ⊥ , RX ⊕ T, with X ∈ – linear subspaces of T (including {0}). ii) The ideals of D are {0}, T and D. C OROLLARY 1.4.– The Lie subalgebras of D which are supplementary of T are the Lie algebras of the form Za with a ∈ E.

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2 Let s be a Lie subalgebra of D, put: A = {ω X | X ∈ s},

u=s∩T

Then A is a Lie subalgebra of (E, ∧) and its dimension is 0, 1 or 3 (lemma 1.6) and u is invariant under the action of ad X for all X ∈ s. S UBALGEBRAS s SUCH THAT dim A = 0. They are the linear subspaces of T (making commutative Lie subalgebras). S UBALGEBRAS s A = Rω X so that

SUCH THAT

dim A = 1. There is X ∈ s (and ∈ / T) such that

s = RX ⊕ u The only possibilities are those of Lemma 1.7. Conversely, when u is one of those spaces s is a Lie subalgebra: if X1 = α1 X + U1 ∈ s, X2 = α2 X + U2 ∈ s, with U1 and U2 ∈ u, since [U1 , U2 ] = 0:   [X1 , X2 ] = [X, α1 U2 − α2 U1 ] ≡ ad X. α1 U2 − α2 U1 ∈ u ⊂ s. S UBALGEBRAS s SUCH THAT dim A = 3. There exist three elements X1 , X2 , X3 of s such that {ω 1 , ω 2 , ω 3 } is a basis of E. According to Lemma 1.7 the three lists: {0},

RXi ,

T ∩ Xi⊥ ,

T for i = 1, 2, 3.

must have in common a space equal to u. Since the dimensions of these linear subspace are respectively 0,1,2,3 for all i, there are only four possibilities: u = {0} or u = T or u = RX1 = RX2 = RX3 or u = T ∩ X1⊥ = T ∩ X2⊥ = T ∩ X3⊥ . Case u = {0} and dim A = 3. The Kernel of the linear map X → ω X (defined on s) is {0}, therefore this map is a Lie algebra isomorphism of s onto (E, ∧). Let us choose (X1 , X2 , X3 ) such that (ω 1 = e1 , ω 2 = e2 , ω 3 = e3 ) be an orthonormal right-handed basis of E; therefore (X1 , X2 , X3 ) is a basis of s and, by isomorphy, for all circular permutation (i, j, k) of (1, 2, 3) we have Xi = [Xj , Xk ] = [Xj , [Xi , Xj ]]. With formula [1.99]: Xi = [Xj , [Xi , Xj ]] = [[Xj | Xj ]]Xi − [[Xj | Xi ]]Xj + (ej · ej )Xi − (ej · ei )Xj = [[Xj | Xj ]]Xi − [[Xj | Xi ]]Xj + Xi . 0 = [[Xj | Xj ]]Xi − [[Xj | Xi ]]Xj .

[1.117]

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79

Since the elements Xi and Xj are linearly independent this relation implies [[Xj | Xj ]] = 0,

[[Xj | Xi ]] = 0.

These relations, holding for all i and j, imply that Xi ∈ Z for i = 1, 2, 3 and that for all pair (i, j) the axes of Xi and Xj lie in the same plane and are perpendicular. Hence the three axes meet at a point a ∈ E and X1 , X2 , X3 ∈ za . Since (X1 , X2 , X3 ) is a basis of s we have s = Za . Case u = T and dim A = 3. Then s contains T and the subspace spanned by {X1 , X2 , X3 } such that (ω 1 , ω 2 , ω 3 ) is a basis of E. It is easy to verify that these subspaces are supplementary. Therefore s = D. I DEALS OF D. According to Lemma 1.6 a subalgebra can be an ideal only when A = {0} or E. When A = {0}, s = u and, according to Lemma 1.7, u can be only {0} or T. When A = {0}, a priori s is of the form D or Za . However since [Za , T] = T and T is not included in Za , the Lie algebra Za is not an ideal. For the Lie algebras of the corollary, necessarily A = E. 1.8.1. Lie subgroups of D A precise mathematical theory of relations between Lie subalgebras of the algebra of a Lie group and Lie subgroups is rather difficult (for more details and references see [CHE 06] Chap IX or [KOB 63] where a slightly different definition of Lie subgroups is used). Here we only wish to show a one-to-one the relation between the Lie subalgebras of D and Lie subgroups of the displacement group D. D EFINITION 1.9.– A Lie subgroup of D is a subgroup which is also a submanifold of D5. With this definition, a Lie subgroup is a Lie group (the operations are differentiable in the meaning of the submanifold) moreover, from the standpoint of topology, a 5 Here, by submanifold we mean, imbedded submanifold. A complete investigation of relations between Lie subalgebras and Lie subgroups, comes up against difficulties with the concept of submanifold: to obtain general theorems, it is necessary to consider also immersed submanifolds (in the treatise [KOB 63] Lie subgroup are immersed submanifolds and they are not necessarily closed, for more details see also [CHE 06]). However, with D one does not meet any problem.

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Lie subgroup is always a closed subset. There is a very important converse property (Cartan Theorem): a subgroup of a Lie group which is closed is always a Lie subgroup (that is a submanifold). So that we could say without practical limitation that a Lie subgroup of D is a subgroup which is also a closed submanifold. T HEOREM 1.17.– If H is a Lie subgroup of D then the tangent space Te H ⊂ Te D is a Lie subalgebra H of D. When H is a normal subgroup, H is an ideal of D and is invariant under the adjoint action of D. D EFINITION 1.10.– The Lie subalgebra H of D defined in Theorem 1.17 is the Lie algebra of the Lie subgroup H 2 A tangent vector of D (resp: of the submaniflod H) at e is the tangent vector of a differentiable curve by e in D (resp. by e and lying on H). According to Proposition 1.11 the elements of Te D are the skew-symmetric vector fields of D. Let U , V be tangent vectors of H at e defined by differentiable maps t → At and t → Bt from R to D(E) with A0 = B0 = e and taking their values in H. Since H is a subgroup, the map t → At .Bt also takes its values in H and defines a tangent vector W of H at e. For all p ∈ E and all fixed λ ∈ R: 

 d = U (p) + V (p) = W (p) At .Bt (p) dt t=0   d = λU (p) Aλt (p) dt t=0

Therefore U +V and λU are tangent vectors of H at e and the tangent vectors of H at e make a linear subspace H of D. Let us prove that H is also a Lie subalgebra of D. For fixed D ∈ H the map t → IntD.Bt is a curve by e on H and (see [1.88]): 

 d = D∗ V IntD.Bt (p) dt t=0

Therefore, when D ∈ H, the linear space H of D is invariant under the action of D∗ . Now, the map t → At ∗ V takes its values in the linear subspace H and, for all p ∈ E, (see [1.86]: 

d   At ∗ V (p) dt

 t=0

= [U, V ](p)

  In other words the derivative of the map t → At ∗ V is [U, V ]. Since a linear subspace of a finite dimension vector space is a closed subset we conclude that [U, V ] ∈ H. When H is a normal subgroup the curve t → IntD.Bt lies in H for all D ∈ D so that D∗ V ∈ H and H in invariant under the adjoint action of D. Now,

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  if t → At ∈ D defines U ∈ D, then At ∗ V ∈ H for all t and we conclude that [U, V ] ∈ H for all U ∈ D, V ∈ H. C OROLLARY 1.5.– If D ∈ D, H is a Lie subgroup of D and H its Lie algebra, then IntD.H is a lie subgroup of D and its Lie algebra is D∗ H. T HEOREM 1.18 (Converse of Theorem 1.17 for the displacement group).– All Lie subalgebra of D is a the Lie algebra of a uniquely defined connected Lie subgroup of D. In general it is not possible to associate a Lie subgroup (as a submanifold of the group) to any Lie subalgebra of the Lie algebra of a Lie group; it may happen that a Lie subalgebra would not be the Lie subalgebra of a (closed) subgroup. Such a phenomenon comes up in very simple groups (torus) but it never comes up in the displacement group and then there is a one-one relation between Lie subalgebras of D and Lie subgroups of D. 1.8.2. Trivial Lie subgroups Those groups correspond to trivial Lie subalgebras {0} and D. Dimension 0 Lie subalgebra Dimension 0 Lie Subgroup Kinematic pair {0} {e} Rigid pair Dimension 6 Lie subalgebra Dimension 6 Lie Subgroup Free pair D D (no effective link) 1.8.3. One-parameter subgroups Those groups are the ranges of exponential mappings s → expD (sX) with various choices of X ∈ D (see [1.83] and [1.96]). The Lie subalgebras are the one-dimension subspaces of D. In the following U is a linear subspace of T. The directions of the constant fields U ∈ U will be denoted by u ∈ E, u = 0. If Δ is a straight line, Tr (Δ) is the group of translations in the direction of Δ, Hel(Δ) is the group of helical displacements about the axis Δ. DIMENSION 1 Subalgebras Subgroups U ⊂ T with dim(U) = 1 Tr (Δ) RZ with Z ∈ / T, [[Z | Z]] = 0 Rot(ΔZ ) RX with X ∈ / T, [[X | X]] = 0 G subgroup of Hel(ΔZ )

Kinematic pairs Prismatic pair (*) Rotary pair (*) Helical pair (*)

In robotics rotary pairs are also called “revolute pairs”.

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1.8.4. Two-parameter subgroups The Lie subalgebras are two dimension subalgebras of D. When U ⊂ T the fields U ∈ U take their constant values u in a two-dimension subspace Π of E; Π, is a plane of direction Π in E, Tr (Π) is the subgroup of the translations in the direction of Π. If Δ is a straight line in E, Hel(Δ) is the subgroup of the helical displacements about the axis Δ in E. DIMENSION 2 Subalgebras Subgroups Kinematic pairs U ⊂ T with dim(U) = 2 Tr (Π) Plane translational pair RX + RX with X ∈ / T, [[X | X]] = 0 Hel(ΔX ) Cylindrical pair (*) In robotics cynlindrical pairs are also called “revolute-slider pairs”. 1.8.5. Three-parameter subgroups The subgroup Pl(Π) is the subgroup of the displacements globally preserving a plane Π which is orthogonal to the axis Δ (rotations about an axis perpendicular to the plane and translations in the direction of the plane). The name “group Y ”) is the notation of J. M. Hervé. The corresponding mechanical devices - only sequences of kinematic pairs - are described in [HER 99] p. 216. DIMENSION 3 Subalgebras Subgroups Kinematic pairs T Tr (E) Translational pairs RX + T ∩ X ⊥ with X ∈ z∗ , [[X | X]] = 0 Pl(Π) Plane pair (*) ⊥ ∗ RX + T ∩ X with X ∈ z , [[X | X]] = 0 “Group Y” (Chain) Zc Rot(c) Spherical pair (*) 1.8.6. Four-parameter subgroups The straight line Δ is perpendicular to the plans Π. The name “group X” is the notation of J. M. Hervé. The many corresponding mechanical devices - only sequences of kinematic pairs - are described in [HER 99] pp. 217–225. DIMENSION 4 Subalgebras Subgroups Kinematic pairs RZ + T with Z ∈ / T Schoenflies group Schoenflies chain pairs  Tr (Π) + Hel(Δ) (“Group X”)

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1.8.7. Five-parameter subgroups No five-dimension Lie subalgebra of D exists, therefore no five-parameter subgroup of D may exist. All the Lie subgroups of D and their various making with mechanical devices (joints) were presented in details by J. M. Hervé [HER 99] pp. 185–236 and [HER 94]. In the former article the motions in the groups are produced by sequences of links allowing motions in one-parameter groups (prismatic, rotary and helical pairs).

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2 Dual Numbers and “Dual Vectors” in Kinematics

The matter at issue in the present chapter is specific of the Euclidean displacement group D(E) of the three-dimension Euclidean oriented affine space E and its Lie algebra D(E) and we refer to Chapter 1, to simplify, from now on they will be respectively denoted by D and D. From the standpoint of real algebra and real analysis, the structures of D and D were studied in details in section 1.5, section 1.5.2 and section 1.5.4; in particular D is a Lie group acting on D by the left action A → A∗ which is nothing but the adjoint action. We now take another standpoint which is based on dual numbers and was introduced by Kotelnikov in Kazan in 1895 (in a today unavailable article), expanded by P.M. Dimentberg [DIM 78] (in Russian) or [DIM 65] (1965, in English), A. T. Yang and F. Freudenstein [YAN 64] (1964), G. R. Veldkamp [VEL 76]. As in section 1.5 and 1.5.2, we follow the coordinatefree presentation introduced in D.P. Chevallier [CHE 91] (1991), and expanded in [CHE 95, CHE 96, CHE 99]. All the computations of kinematics of rigid body systems may be expressed in the dual number and “dual vectors” setting and this stand point brings much simplicity in the formal aspects of those computations. However, dynamics introduces operations which are not in agreement with this setting (clearly: kinematics deals with operations turning out to be to be all Δ-linear, whereas dynamics deals with some operations which are not Δ-linear). Therefore, in practical applications, say to design softwares, the kinematic calculations may be performed in this setting (with the structure of a Δmodule D, in dimension 3 over Δ) while, for the calculations regarding dynamics, it is necessary to switch to the mathematical expression with the real vector space D (in dimension 6 over R).

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2.1. The Euclidean module D over the dual number ring 2.1.1. The ring Δ and the module structure on D Dual numbers are “numbers” expressed by z = a + b, with a and b in R, and  satisfying 2 = 0 such that z = a + b and z  = a + b are equal if and only if a = a and b = b . Dual numbers make an abelian ring (see. [VEL 76, CHE 99, BOT 90a]) denoted by Δ. Operations in the ring Δ verify: (a + b) + (a + b ) = (a + a ) + (b + b ),

(a + b).(a + b )

= aa + (ab + a b), The zero and unity of Δ are 0 = 0 + 0, 1 = 1 + 0. There is a conjugation in the ring Δ such that z = a − b when z = a + b. Obviously z = z (the conjugation is involute), zz = zz = a2 and: z1 + z2 = z 1 + z 2 ,

z1 .z2 = z 1 .z 2 .

In an obvious meaning R is a subring of Δ. There are divisors of zero in this ring and it is easy to prove that they are the members of Δ which are of the form b with b ∈ R. R EMARK 2.1.– A mathematician would consider that the above definition is fuzzy; a more precise definition, similar to that of complex numbers, could say that Δ is a subset of R × R endowed with two laws defined by (a, a ) + (b, b ) = (a + a , b + b ),

(a, a ).(b, b ) = (aa , ab + a b).

Then 0 ≡ (0, 0), 1 ≡ (1, 0) and  = (0, 1) and the conjugation reads (a, b) = (a, −b). Recall that a module over an abelian ring with unity, say Δ, is a set M endowed with an abelian group operation (+) and an external law (λ, x) ; λx, with λ ∈ Δ and x ∈ M. The axioms of a Δ-module are similar to those of a vector space, except that scalars are picked in a ring, say Δ, not in a field. Let us note that, in spite of this seemingly slight difference, the properties of vector spaces and modules might be quite different (see for example [BIR 67]). In the following theorem we note O the vector field p → 0 (the null vector of E) and −X the vectorfield p → −X(p) (the opposite of X(p) in E), that is to say the vectors O and −X of the vector space D over R.

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T HEOREM 2.1.– The space D has a natural Δ-module structure extending its real vector space structure i.e.: for X, Y and Z ∈ D, λ and μ ∈ Δ:

i)

⎧ X + Y = Y + X, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ X + (Y + Z) = (X + Y ) + Z, ⎪ ⎪ O + X = X + O = X, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ X + (−X) = (−X) + X,

ii)

⎧ λ.(X + Y ) = λ.X + λ.Y, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ (λ + μ).X = λ.X + μ.X, ⎪ ⎪ λ.(μ.X) = (λμ).X, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1.X = X.

and D is also a Lie algebra over Δ, i.e.: iii)

[λ.X, Y ] = λ.[X, Y ],

iv)

[[λ.X | Y ]] = [[X | λ.Y ]].

(relation (iii) implies [X, λ.Y ] = [λ.X, Y ] = λ.[X, Y ] and that the Lie bracket (X, Y ) → [X, Y ] is a skew-symmetric Δ-bilinear map, however to interpret (iv) is not so simple). 2 Relations (i) are those which are valid in the (abelian group of the) vector space D over R. In order that operation (λ, X) → λ.X makes sens when λ ∈ Δ it suffices to define .X for all X ∈ D; we put: .X = the constant vector field equal to ω X on E. Hence .X ∈ T and .(.X) = 0. In other words, the product of the scalar  ∈ Δ by X ∈ D for the law of the module is defined as the action of a “linear operator ” : D → D with range T and vanishing square defined in section 1.5.6. This definition extends to any general λ = a + b ∈ Δ and X ∈ D; the vector field λ.X will be: λ.X = aX +  b X : p → aX(p) + b ω X

[2.1]

This formula is consistent for ω bX = bω X when b ∈ R and bX (product of X by b for the law of the real vector space D) equal  b X (product of b X by  for the so defined product by the dual number ). The checking of properties (ii) is now straightforward. For the Lie bracket and the Klein form we have .[X, Y ] = [.X, Y ] = [X, .Y ] = constant field equal to ω X ∧ ω Y , [[.X | Y ]] = [[X | .Y ]] = ω X · ω Y , according to the definition of .X and to formula [1.71] for the Klein form. The checking of properties (iii) and (iv) is then straightforward.

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2.1.2. Linear independence over Δ With Theorem 2.1, D is endowed with two distinct structures: six dimension vector space over R and module over Δ (which is three dimension, see Proposition 2.2). It is necessary to distinguish carefully one of those structures from the other, in particular when properties as linear independence or linearity are involved (sections 2.1.2 and 2.1.3). With the two structures defined on D two kinds of linear independence in D (over R or over Δ) may be derived according to the choice of the coefficients of the linear combinations. The links between these two kinds of linear independence and the help from the dual number method in the testing of linear independence over R were exposed in detail in article [CHE 95] (1995). P ROPOSITION 2.1.– Let X1 , . . . , Xn be members of D. The following properties are equivalent: i) X1 , . . . , Xn are linearly independent over Δ. ii) X1 , . . . , Xn are linearly independent over R. iii) ω 1 , . . . , ω n are linearly independent over R (in E). (It is evident that these properties may be true only when n ≤ 3). 2 i) ⇒ ii). A relation of the form μ1 X1 +· · ·+μn Xn = 0, with μ1 , . . . , μn ∈ R is the vanishing of a linear combination of X1 , . . . , Xn with coefficients μ1 , . . . , μn ∈ Δ. Hence μ1 = . . . = μn = 0. ii) ⇔ iii). As vector fields on E, X1 , . . . , Xn are constant vector fields equal to ω 1 , . . . , ω n and these vector fields are linearly independent over R if and only if ω 1 , . . . , ω n are linearly independent in E. ii) ⇒ i). Assume that z1 X1 + · · · + zn Xn = 0 with zi = λi + μi , λi ∈ R, μi ∈ R. Then    z1 X1 + · · · + zn Xn = λ1 X1 + · · · + λn Xn = λ1 (X1 ) + · · · + λn (Xn ) = 0 so that λ1 = · · · = λn = 0 and the relation amounts to μ1 X1 + · · · + μn Xn = μ1 (X1 ) + · · · + μn (Xn ) = 0, so that μ1 , . . . , μn = 0. In the end z1 = . . . = zn = 0.

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P ROPOSITION 2.2 (Basis of the module D).– If X1 , X2 , X3 ∈ D are linearly independent over Δ then (X1 , X2 , X3 ) is a basis of the module D. In other words for every X ∈ D there are uniquely defined z1 , z2 , z3 ∈ Δ such that X = z1 X1 + z2 X2 + z3 X3 . So, as Δ-module, D is dimension 3 over Δ while its dimension is 6 as R-vector space. 2 Proposition 2.1 iii) implies that (ω 1 , ω 2 , ω 3 ) is a basis of E and it is readily deduced that (X1 , X2 , X3 ) is a basis of T. Let X ∈ D, there exist uniquely defined λ1 , λ2 , λ3 ∈ R such that X = λX1 + λ2 X2 + λ3 X3 . With this choice Y = X − (λ1 X1 + λ2 X2 + λ3 X3 ) ∈ T (Y = 0, hence ω Y = 0). Therefore, there exist uniquely defined μ1 , μ2 , μ3 ∈ R such that Y = μ1 X1 + μ2 X2 + μ3 X3 and X = Y + λ1 X1 + λ2 X2 + λ3 X3 = (λ1 + μ1 )X1 + (λ2 + μ2 )X2 + (λ3 + μ3 )X3 In the end X = z1 X1 + z2 X2 + z3 X3 with uniquely defined zi = λi + μi ∈ Δ. The bases of the module D over Δ will be studied with more details in exercise 2.3.

2.1.3. Δ-linear maps The following lemma may be used to simplify reasonings: L EMMA 2.1.– Let M be a Δ-module and f : D → M. The following properties are equivalent: a) f is Δ-linear, b) f is R-linear and verifies f (.X) = .f (X) for all X ∈ D,2 a) ⇒ b) because b) is a particular case of a) compelling the scalars to R or to . b) ⇒ a): f (X + Y ) = f (X) + f (Y ) is verified in the abelian groups D and M and it remains to prove that f (λ.X) = λ.f (X) if λ = a + b, a, b ∈ R.

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f (λX) = f (aX) + f (bX) (R-linearity) = af (X) + f (bX) (second assumption b)) = af (X) + b f (X)

(R-linearity))

= (a + b).f (X) = λ.f (X)

(laws in the module M.)

P ROPOSITION 2.3.– For A ∈ D the map A∗ is Δ-linear (and is an automorphism of the module D). In other words, the operators Ad A deduced from the adjoint action of the group D on its Lie algebra are Δ-linear operators. 2 It is sufficient to prove that A∗ (X) = A∗ X for A ∈ D, X ∈ D and this was proved in section 1.5.6. 2.1.4. Dual inner and mixed products In the module D, the Lie bracket, which not only is a R-bilinear map but also is Δ-linear will play the role of the vector product of the Euclidean vector space E. That module structure of D also allows generalizations of the inner and mixed products and, further, of formulas of ordinary vector algebra in E. Finally there are many structural analogies between the three-dimension real vector space E over R and the three-dimension module over Δ. P ROPOSITION 2.4.– The map defined by (X, Y ) → {X | Y } = ω X · ω Y + [[X | Y ]],

[2.2]

taking its values in Δ, is a symmetric non-degenerate Δ-bilinear form on the module D. The map   (X, Y, Z) → {X; Y ; Z} = {X | [Y, Z]} ≡ ω X ; ω Y ; ω Z + [[X | [Y, Z]]]. is a skew-symmetric Δ-multilinear form of degree 3. These forms are invariant by action of D on D: {A∗ X | A∗ Y } = {X | Y },

{A∗ X; A∗ Y ; A∗ Z}

= {X; Y ; Z} for all A ∈ D, X, Y, Z ∈ D. 2 It is evident that the map X → {X | Y } (with fixed Y ) is R-linear. Since ω X = 0 we have {.X | Y } = [[.X | Y ]] =  ω X · ω Y = {X | Y }

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and this map is Δ-linear by Lemma 2.1. So for the map Y → {X | Y } for fixed X. Hence {· | ·} is Δ-bilinear (and obviously symmetric). Relation {X | Y } = 0 for all Y ∈ D implies [[X | Y ]] = 0 for all Y what implies X = 0 because the Klein form is non-degenerate. Invariance by action of D is a consequence of [1.94] and [1.98] from section 1.5.5 and 1.5.6. With the definition and the invariance of {· | ·}, it is evident that the map (X, Y, Z) → {X; Y ; Z} is a Δ-multilinear form on D and is invariant. The real  part of {X | [Y, Z]} and {Y | [Z, X]} are equal because ω · ω ∧ ωZ = X Y    ω Y · ω Z ∧ ω X = ω X ; ω Y ; ω Z . The dual parts [[X | [Y, Z] ]] and [[Y | [Z, X] ]] are equal due to the properties of the Klein form (see [1.80] and [1.95]). D EFINITION 2.1.– product on D.

- The Δ-bilinear map (X, Y ) → {X | Y } is the dual inner

- The R-bilinear map “real part of the dual inner product” (X, Y ) → (X | Y ) = ω X · ω Y is the Killing form on D. Let us note relation (X | Y ) = [[.X | Y ]] = [[X | .Y ]]. (However the dot product has been defined on E × E whereas the Killing form in defined on D × D.) - The Δ-multilinear map (X, Y, Z) → {X; Y ; Z} (with semi-colons) is the Δmixed product on D. Relation implying that the map (X, Y, Z) → {X; Y ; Z} is skew-symmetric and which has just been proved is an analogy with ordinary vector algebra: {X | [Y, Z]} = {Y | [Z, X]}.

[2.3]

Another analogy, is a generalized Gibbs formula: [X, [Y, Z]] = {X | Z}Y − {X | Y }Z.

[2.4]

2 Let us prove [(2.4]. Put U = [X, [Y, Z]] then, for p ∈ E: U (p) = ω X ∧ [Y, Z](p) − ω [Y,Z] ∧ X(p)     = ω X ∧ ω Y ∧ Z(p) − ω Z ∧ Y (p) − ω Y ∧ ω Z ∧ X(p)         = ω X · Z(p) ω Y − ω X · ω Y Z(p) − ω X · Y (p) ω Z + ω X · ω Z Y (p)     + X(p)· ω Z ω Y − X(p)· ω Y ω Z = [[X | Z]]ω Y − [[X | Y ]]ω Z + ω X · ω Z Y (p) − ω X · ω Y Z(p)     = (ω X · ω Z + [[X | Z]]) Y (p) − (ω X · ω Y + [[X | Y ]]) Z (p)

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and we recognize the value at p of the right hand side of [2.4]. {X | Y }2 = {X | X}{Y | Y } − {[X, Y ] | [X, Y ]}, 2

{X; Y ; Z} = {[X, Y ]; [Y, Z]; [Z, X]},

[2.5] [2.6]

4 {X | Y } = {X + Y | X + Y } − {X − Y | X − Y }.

[2.7]

A “dual norm” N is derived from the dual inner product (for a while we anticipate section 2.4.2 and use the square root of dual numbers which is defined when their real part is positive or null). We put N (X) =



{X | X},

and relation [2.7] may take the form of a well-known relation of classical vector algebra: {X | Y } =

1 1 N (X + Y )2 − N (X − Y )2 . 4 4

[2.8]

However, for this “norm”, N (X) = 0 is not equivalent to X = 0. Let Zp = {X ∈ D | X(p) = 0},

Z=



Zp ,

Z∗ = {X ∈ Z | X = 0},

p∈E

then, more precisely: / T, N (X) = ω X (1 + fX ) whenX ∈

[2.9]

N (X) = 0 ⇐⇒ {X | X} = 0 ⇐⇒ X ∈ T, N (X) ∈ R∗+ ⇐⇒ {X | X} ∈ R∗ ⇐⇒ X ∈ Z∗ . With every element X ∈ D − T is associated a normalized element υ such that X = N (X) υ with N (υ) = 1, actually, υ = N (X)−1 X = ω X −1 (1 − fX )X, and the splitting of X ∈ / T of Theorem 1.14 is expressed by: X = Z + U with Z = ω X υ,

U = fX X.

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An element X of Z∗ has a well-defined axis (ΔX , ω X ) in E, i.e. a straight line oriented by the director ω X . A normalized element υ of D (i.e. such that {υ | υ} = 1) defines an axis (ΔU , u) with a normalized director ω U = u. Therefore the quadric Q = {υ ∈ D | {υ | υ} = 1} = {υ ∈ Z∗ | N (υ) = 1} may be considered as “the space of oriented straight lines of E ”. When a line Δ is given, the two axes on Δ with opposite orientations on Δ correspond to the normalized elements υ and −υ of Q such that if c ∈ Δ and u ∈ E is a normalized director: → υ(p) = u ∧ − cp,

→ −υ(p) = (−u) ∧ − cp.

The geometrical meaning of the cancellation of the dual inner product and the dual mixed product is pointed out in the following proposition: P ROPOSITION 2.5.– Let X, Y and Z be in D. – If X, Y and Z ∈ D − T. Then {X | Y } = 0 ⇐⇒ ΔX and ΔY are in the same plane and perpendicular.

{X; Y ; Z} = 0 ⇐⇒

either ΔX , ΔY and ΔZ have a common perpendicular or ΔX , ΔY and ΔZ are parallel.

(in other words ΔX , ΔY and ΔZ meet a straight line at right or are parallel). – If X ∈ T and Y ∈ D − T. Then {X | Y } = 0 ⇐⇒ the direction of X and the direction of ΔY are orthogonal. – If X ∈ T and Y ∈ T then {X | Y } = 0 (the space T is isotropic for the dual inner product). 2 When {X | Y } = 0 the scalar product ω X · ω Y vanishes and the axes are perpendicular and in the same plane (Proposition 1.12). Conversely, when this property is verified, {X | Y } = 0. When {X; Y ; Z} = 0 the mixed product (ω X ; ω Y ; ω Z ) vanishes and the rank of (ω X , ω Y , ω Z ) is either 1 (and then the axes ΔX , ΔY and ΔZ are parallel) or 2. In the latter case one of the Lie brackets [X, Y ], [Y, Z], [Z, X] is not in T. If, for example [X, Y ] ∈ / T, its axis is the common perpendicular to ΔX and ΔY (exercise 1.35) and this perpendicular meets ΔZ because {[X, Y ], Z} = {X; Y ; Z} = 0. It is easy to check that conversely {X; Y ; Z} = 0 when the axes ΔX , ΔY and ΔZ are parallel or when they have a common perpendicular.

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If (X1 , X2 , X3 ) is a basis of the module D then (X1 ; X2 ; X3 ) = (ω 1 ; ω 2 ; ω 3 ) = 0 according to Proposition 2.2. Therefore, as those of E, the bases of D share out among two classes respectively called right–handed bases (for which

(X1 ; X2 ; X3 ) > 0) and left-hand bases (for which (X1 ; X2 ; X3 ) < 0). If X1 , X2 , X3 are such that {Xi | Xi } = 1, {Xi | Xj } = 0 for i = j then (ω 1 , ω 2 , ω 3 ) is an orthonormal basis of E, and then (X1 , X2 , X3 ) is a basis of D over Δ (Propositions 2.1 and 2.2). Such a basis will be called an orthonormal basis of (the module) D. When (X1 , X2 , X3 ) is an orthonormal basis of D, the axes Δ1 , Δ2 , Δ3 (oriented by ω 1 , ω 2 , ω 3 ) are orthogonal axes in E meeting at a point c ∈ E so that they define an orthonormal affine frame of E 1. Conversely if (c; ω 1 , ω 2 , ω 3 ) is an orthonormal affine frame in E there exist a unique orthonormal basis (X1 , X2 , X3 ) of D such that the axes of X1 , X2 , X3 are respectively the lines by c oriented by ω 1 , ω 2 , ω 3 . In other words there is a bijective relation between the orthonormal frames of D and the orthonormal affine frames of E. In the same manner we see that there is a bijective relation between the orthonormal right-handed frames of D and the orthonormal right-handed affine frames of E. E XERCISE 2.1.– Let λ = a + b be an invertible dual number (a = 0) and X ∈ D − T. Prove that the axes of X and λX are identical and that, with the notation of Theorem 1.12 of Chapter 1: fλX = fX + b. E XERCISE 2.2.– (Complement to exercise 1.36-1).) Prove that when X, Y and [X, Y ] are not in T then the axis of [X, Y ] is the common perpendicular to ΔX and ΔY . H INT.– Use Proposition 2.5 and that (X, Y, Z) → {X; Y ; Z} is skew-symmetric. E XERCISE 2.3.– Let (X1 , X2 , X3 ) be a basis of D. 1) Prove that if the basis is orthonormal, then the dual coordinates of Y (such that Y = ξ1 X1 + ξ2 X2 + ξ3 X3 ) are expressed by ξk = {Xk | Y } for k = 1, 2, 3. 2) Prove that, in general when the basis is not orthonormal, the dual coordinates of Y are expressed by: 1 1 {Y ; X2 ; X3 }, ξ2 = {Y ; X3 ; X1 }, H H 1 ξ3 = {Y ; X1 ; X2 } with H = {X1 ; X2 ; X3 }. H ξ1 =

1 See Proposition 1.12: Δ1 and Δ2 lie in the same plane orthogonal to Δ3 and they meet at a point c, Δ3 meet this plane at a point which is necessarily c.

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3) Prove that the following properties are equivalent: i) (X1 , X2 , X3 ) is a basis of the Δ-module D, ii) {X  1 ; X2 ; X3 } is an invertible dual number, iii) [X2 , X3 ], [X3 , X2 ], [X3 , X1 ] is a basis of the Δ-module D, iv) (ω 1 , ω 2 , ω 3 ) is a linearly independent family in E.

[2.10]

  4) Check that the expansion of Y in the basis [X2 , X3 ], [X3 , X2 ], [X3 , X1 ] reads Y = μ1 [X2 , X3 ] + μ2 [X3 , X1 ] + μ3 [X1 , X2 ] with μ1 =

{X1 | Y } , {X1 ; X2 ; X3 }

μ2 =

{X2 | Y } {X1 ; X2 ; X3 }

μ3 =

{X3 | Y } {X1 ; X2 ; X3 }

E XERCISE 2.4.– Let X and Y ∈ T, put Z = [X, Y ]. Prove that the following properties are equivalent: i) X and Y are linearly independent over Δ, ii) [X, Y ] ∈ / T, iii) The set (X, Y, Z) is a basis of D over Δ. In particular: X and Y are linearly dependent over Δ ⇐⇒ [X, Y ] ∈ T. E XERCISE 2.5.– Let us define the mapping M : D × D × D → D by M(X, Y, Z) = {X; Y ; Z}[Y, Z]. Recall that two elements of D are said to be reciprocal when they are orthogonal with respect to the Klein form. 1) Prove that M(X, Y, Z) is reciprocal to X, Y and Z. 

2) Prove that if (X, Y, Z) is a  basis of the Δ module D, M(X, Y, Z), M(Y, Z, X), M(Z, X, Y ) is also a basis.

then

3) Prove that a necessary and sufficient condition for U be in the linear subspace spanned by X, Y, Z in the vector space D over R is that U be reciprocal to M(X, Y, Z), M(Y, Z, X), M(Z, X, Y ). H INT.– 1) Remark that [[X | M(X, Y, Z)]] is a real number and see [2.3]. 2) See exercise 2.2. 3) Let V and V⊥ be the linear subspace generated by X, Y, Z in the six-dimension vector space D over R and its orthogonal for the Klein form. Then   V = V⊥ ⊥ . Use question 2.

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E XERCISE 2.6.– Let S = (X1 , X2 , X3 , X4 ) be a system of members of D such that (ω 1 , ω 2 , ω 3 ) be a basis of E. 1) Prove that for all choice of x1 , x2 , x3 , x4 in the Euclidean vector space E: (x2 ; x3 ; x4 ) x1 + (x3 ; x1 ; x4 ) x2 + (x1 ; x2 ; x4 ) x3 − (x1 ; x2 ; x3 ) x4 = 0 () 2) Prove that a necessary and sufficient condition for the rank of S be equal to 3 is that the following three relations hold: ⎧ ⎨ (ω 1 ; ω 2 ; ω 3 )[[X2 ; X3 ; X4 ]] − (ω 2 ; ω 3 ; ω 4 )[[X1 ; X2 ; X3 ]] = 0 (ω 1 ; ω 2 ; ω 3 )[[X3 ; X1 ; X4 ]] − (ω 3 ; ω 1 ; ω 4 )[[X1 ; X2 ; X3 ]] = 0 ⎩ (ω 1 ; ω 2 ; ω 3 )[[X1 ; X2 ; X4 ]] − (ω 1 ; ω 2 ; ω 4 )[[X1 ; X2 ; X3 ]] = 0 and then real numbers α1 , α2 , α3 , α4 , not all equal 0, such that α1 X1 + α2 X2 + α3 X3 + α4 X4 = 0 are of the form α1 = k

(ω 2 ; ω 3 ; ω 4 ) , (ω 1 ; ω 2 ; ω 3 )

α2 = k

(ω 3 ; ω 1 ; ω 4 ) , (ω 1 ; ω 2 ; ω 3 )

α3 = k

(ω 1 ; ω 2 ; ω 4 ) , (ω 1 ; ω 2 ; ω 3 )

α4 = −k

with k ∈ R, k = 0. 3) Let us put VS = {X2 ; X3 ; X4 } X1 + {X3 ; X1 ; X4 } X2 +{X1 ; X2 ; X4 } X3 − {X1 ; X2 ; X3 } X4 US

= (ω 2 ; ω 3 ; ω 4 ) X1 + (ω 3 ; ω 1 ; ω 4 ) X2 +(ω 1 ; ω 2 ; ω 4 ) X3 − (ω 1 ; ω 2 ; ω 3 ) X4

Then i) For all choice of the elements Xi ∈ D relations VS = 0 and US ∈ T are verified. ii) When the rank of S over Δ equal 3, a necessary and sufficient condition for the rank of S over R be also equal 3 is that US = 0. H INT.– For question 2, (X1 , X2 , X3 ) is a basis of the Δ-module D. The element X4 has a uniquely defined expansion, with dual coefficients, in this basis.

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2.2. Dualization of a real vector space 2.2.1. General extension of a real vector space into a Δ-module Starting from a real vector space (a R−module), a standard algebraic process, i.e.  with the extension of the scalar ring of a module, leads to an extended Δ-module E the following properties:  – There exists a natural isomorphism from E onto a real vector subspace of E. – Every R-linear map u from E into a Δ-module M extends in a unique way as  to M such that u a Δ-linear map u ˆ from E ˆ(x) = u(x) for x ∈ E (identified with a  subset of E).  is not unique (namely, it is defined up to an isomorphism The extended module E of modules) and thus for a particular application one may chose the most convenient  as the tensor product Δ⊗E module to play this role. A mathematician would define E of two vector spaces over R and prove that it has a natural Δ-module structure. For  from E: let E × E be sake of clarity, we explain a simpler construction of a module E the cartesian product of E by itself which is a vector space over R for the operations defined by: (x, y) + (x , y ) = (x + x , y + y ),

[2.11]

α(x, y) = (αx, αy) for α ∈ R.

[2.12]

Now, if we put λ(x, y) = (αx, αy + βx) for λ = α + β ∈ Δ,

[2.13]

it is readily proved that, endowed with the addition [2.11] and the multiplication by  as sets E × E and E  are identical, scalars [2.13], E × E is a Δ-module denoted by E;  they differ about the algebraic structures. Let us prove that E possesses the required  contains two complementary vector subspaces properties for an extended module. E over the real field: {(x, 0) | x ∈ E} = E × {0},

{(0, y) | y ∈ E} = {0} × E.

If the elements x of E and (x, 0) of E × {0} are identified, then E itself appears as  Of course E × {0} is not a submodule since, for example, a real vector subspace of E. (x, 0) = (0, x), a relation justifying the isomorphism {0} × E  E. According to these conventions, we may write the direct sum of vector subspaces over the real field:  = E ⊕ E, E

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 are dual vectors, the vectors in E (respectively E) are the real The vectors in E  Thus, every element z of E  may be vectors (respectively pure dual vectors) of E. expressed as the sum of a real and a pure dual vector according to the more common notation z = (x, y) = x + y. Let M be a Δ-module and let u : E → M be a R-linear map, put  u ˆ(x + y) = u(x) + u(y) for x + y ∈ E.

[2.14]

Since u ˆ satisfies condition (b) of Lemma 2.1, u ˆ is a Δ-linear map, and with the  identification of E to a subset of E, u ˆ(x) = u(x) for x ∈ E. Moreover, it is readily checked that u ˆ defined by [2.14] is the sole extension of u which has these properties.  and F  be the extended modules of vector spaces E and F. E XERCISE 2.7.– 1) Let E  to F  is (compare [2.14] and Prove that the general form of a Δ-linear map from E [2.15]): w(x + y) = u(x) + [v(x) + u(y)] (in other words w(x, y) = (u(x), v(x) + u(y))).

[2.15]

with u and v linear.  be the Δ-module of Δ-linear operators in E  and let L  be the Δ2) Let LΔ (E) module L ⊕ L constructed by the extension process applied to the vector space L = L(E) of linear operators on E. Check that the map (u, v) → w defined in 2.15 is an  onto L (E).  isomorphism of Δ-modules from L Δ Let us explain the matrixform of the  results proved in exercise 2.7 for finite ˆi = (ei , 0), dimensional vector spaces. If e1 , . . . , en is a basis of E then, putting e    ˆ1 , . . . , e ˆn is a basis of E and every dual vector expresses as e z=

n

ˆi , zi e

i=1

    with uniquely defined dual numbers zi . If 1 , . . . ,p and ˆ 1 , . . . , ˆ p are  then in the setting of [2.15], the Δ-linear map w corresponding bases in F and F, is described by a dual matrix ⎤ ⎡ ⎡ ⎤ ⎤ ⎡ w11 . . . w1n u11 . . . u1n v11 . . . v1n Mat(w) = ⎣ . . . . . . . . . ⎦ = ⎣ . . . . . . . . . ⎦ +  ⎣ . . . . . . . . . ⎦ , wp1 . . . wpn up1 . . . upn vp1 . . . vpn where the real matrices [uij ] and [vij ] are those of the linear maps u and v.

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  L (E)  is merely In the finite dimensional framework, the isomorphism L Δ the relation transforming the pair of real matrices ([uij ], [vij ]) into the dual matrix  in the setting of [wij ] = [uij ]+[vij ] according to the previous formula. When M = F,       ˆ ˆn and 1 , . . . , ˆ p , the extended ˆ1 , . . . , e formula [2.14], with respect to the bases e map u ˆ is described by a real matrix ⎤ ⎡ u11 . . . u1n Mat(ˆ u) = ⎣ . . . . . . . . . ⎦ up1 . . . upn     which is merely the matrix of u relative to the bases e1 , . . . , en and 1 , . . . ,p .  the matrix of u However, if M is not an extended vector space F, ˆ is not a real matrix in general. 2.2.2. Dualization of the Euclidean vector space in dimension 3  is When E is the Euclidean vector space in dimension 3 the associate Δ-module E  the set of “dual vectors” used in kinematics. The operations in E may be extended to E into similar Δ-linear or Δ-multilinear operations. If we denote by x1 · x2 , x1 ∧ x2 and (x1 ; x2 ; x3 ) the dot product, the vector product and the mixed product in E, using  if zi = xi + yi (i = 1, 2, 3): similar notation in E,   z1 · z2 = x1 · x2 +  y1 · x2 + x1 · y2   z1 ∧ z2 = x1 ∧ x2 +  y1 ∧ x2 + x1 ∧ y2   (z1 ; z2 ; z3 ) = (x1 ; x2 ; x3 ) +  (y1 ; x2 ; x3 ) + (x1 ; y2 ; x3 ) + (x1 ; x2 ; y3 )        When e1 , e2 , e3 ˆ2 , e ˆ3 is a basis of E. ˆ1 , e If e1 , e2 , e3 is a basis of E then  e  ˆ2 , e ˆ3 is orthonormal according to the ˆ1 , e is orthonormal (resp. right-handed) e ˆj = e ˆk when (i, j, k) is a circular permutation ˆi ∧ e extended inner product resp. and e   Moreover, if the coordinates of dual vectors x ˆ, y ˆ are denoted by of (1, 2, 3) in E. x1 , . . . , y3 their extended dual products take the common forms: ˆ·y ˆ = x1 y1 + x2 y2 + x3 y3 , x    ˆ1 + x3 y1 − x1 y3 ) e ˆ2 + x1 y2 − x2 y1 ) e ˆ3 . ˆ∧y ˆ = x2 y3 − x3 y2 ) e x Let us turn to the relation between dual vectors and “screws” (“torsors”, “motors”...) describable in D and used in mechanics. For fixed o ∈ E (an origin in E)  be the map let Jo : D → E Jo : X → ω X + X(o).

[2.16]

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Note that, according to the classical terminology, the real and dual parts of the dual vector Jo (X) are the Plücker vectors at o associated to X. P ROPOSITION 2.6.– For fixed o ∈ E (origin) the map Jo is an isomorphism of Δ and for X, Y and Z ∈ D: modules from D onto E ⎧  ⎨ Jo [X, Y ]) = Jo (X) ∧ Jo (Y ), {X | Y } = J  o (X) · Jo (Y ),  ⎩ {X; Y ; Z} = Jo (X); Jo (Y ); Jo (Z) ,

[2.17]

(where to the right there are the extended operations of E.) C OROLLARY 2.1.– Jo is an isomorphism of euclidean dual-modules D endowed with  endowed with the extended dot product. the dual inner product onto E 2 It is clear that Jo (X +Y ) = Jo (X)+Jo (Y ). If λ = α+β, applying the definitions we find on the one hand (definition of Jo ) λX = αX + βX and Jo (λX) = αω X + (αX(o) + βω X ), on the other hand (definition (2.13)) λJo (X) = (α + β)(ω X + X(o)) = αω X + (αX(o) + βω X ).   Hence Jo (λX) = λJo (X). Let e1 , e2 , e3 be an orthonormal  basis of E and o ∈ E and if we define ξ 1 , ξ 2 , ξ3 ∈ D by ξ i (o) = 0, ω i = ei , then ξ1 , ξ 2 , ξ 3 is a basis  ˆi = of the module D, orthogonal for the dual inner product. Moreover, if Jo (ξ i ) = e     for the extendd ˆ1 , e ˆ2 , e ˆ3 is an orthonormal basis of E (0, ei ) for i = 1, 2, 3 where e dot-product. Since Jo is Δ-linear and transformed a basis of the module D into a basis  it is an isomorphism of Δ-modules. The proof of formula [2.17] is of the module E, the matter of exercise 2.8.   The axes of the ξ i are those of the orthogonal frame o; e1 , e2 , e3 of E. The proof uses a correspondence between orthogonal frames of E with origin o, orthogonal bases  and orthogonal bases of D. of E After all, calculations on the elements of D are equivalent to calculations on dual vectors. However the latter depends on the choice of an origin in space E whereas the former are intrinsic and generally they take a more compact form. E XERCISE 2.8.– Check that for fixed o ∈ E (origin) formula [2.17] are verified.  is said to be skew-symmetric if E XERCISE 2.9.– A Δ-linear operator w ∈ LΔ (E)  z1 · w(z2 ) + w(z1 ) · z2 = 0 for all z1 and z2 in E.

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1) Prove that when w is a skew-symmetric operator there exists a uniquely defined  such that w(z) = a ∧ z for all z ∈ E  and with the extended vector-product. a∈E ∼

∼

2) Let a denote the map z → a ∧ z. Prove that the map a → a is an isomorphism  onto the submodule LaΔ (E)  of L (E).  of Δ modules of E Δ 3) Let La (E) be the vector space of the skew-symmetric operators in the vector  space E. Prove that the Δ-module L a (E), constructed by the dualization process  applied to the vector space La (E), is isomorphic to the submodule LaΔ (E). E XERCISE 2.10.– 1) Let E be the three-dimension Euclidean vector space and ϕ be a symmetric bilinear form on E. Prove that ϕ is invariant under the action of SO(E) if and only if ϕ is proportional to the dot product (i.e. there exists k ∈ R such that ϕ(x, y) = k x · y). 2) A symmetric R-bilinear form on D is said to be invariant under the action of D on D when Φ(A∗ X, A∗ Y ) = Φ(X, Y ) for all A in D, X, Y in D. Prove that Φ is invariant under the action of D if and only if there exist real numbers a and b such that Φ(X, Y ) = a(X | Y ) + b[[X | Y ]] for all X and Y in D. 3) Check that Φ is non-degenerate if and only if b = 0. 4) Prove that there is no symmetric positive definite bilinear form invariant by the action of D on D. 5) Prove that a symmetric Δ-bilinear form Ψ on D is invariant under the action of D on D if and only if there exists a dual number d such that Ψ(X, Y ) = d{X | Y } for all X and Y in D. H INT.– 1) Use an orthonormal basis of E. 2) Translate the invariance property by the   isomorphism X → ω X , X(o) . A proof of those results was exposed in [CHE 99], pp. 122–125. The negative answer to question 4, exercise 2.10, shows that it is not possible to define by a natural way a measure of wrenches or screws agreeing with the invariance properties of mechanics. This problem was discussed by J. Duffy in [DUF 90] and [DUF 96]. 2.2.3. The groups O(D) and SO(D) The properties of the structure of D endowed with the dual inner product and the dual mixed product are very close to the properties of an Euclidean vector space except

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that the scalars are picked in Δ rather than in R. Therefore the following definitions are natural. D EFINITION 2.2 (Orthogonal and special orthogonal groups of D).– – O(D) is the set of the maps f : D → D preserving the dual inner product, i.e. such that for all X and Y in D: {f (X) | f (Y )} = {X | Y }. – SO(D) is the set of the maps f : D → D preserving the dual inner product and the dual mixed product, i.e. the maps such that for all X, Y and Z in D: {f (X) | f (Y )} = {X | Y },

{f (X); f (Y ); f (Z)} = {X; Y ; Z}.

Note that neither the Δ-linearity nor the R-linearity of f is assumed in this definition; Δ-linearity will be a consequence of the definition. The following Theorem characterizes the group O(D): T HEOREM 2.2.– Let f be a mapping from D to D. The following properties are equivalent: i) f ∈ O(D), ii) f is Δ-linear and maps all orthonormal basis of D into an orthonormal basis of D. iii) f is Δ-linear and there is an orthonormal basis of D mapped by f into an orthonormal basis of D. iv) f is Δ-linear and preserves the dual norm, that is to say: N (f (X)) = N (X) for all X in D. v) f is Δ-linear and there is A ∈ D such that either f = A∗ or f = −A∗ . Under thoses conditions the element A ∈ D of v) is uniquely defined.   Since A∗ ∈ L(D), −A∗ = − A∗ is meaningful however, if A is an affine map −A has no meaning. The map A → A∗ from D to the linear group Gl(D) is one-one (because A∗ = 1 if and only if A =identity) therefore the subset D∗ = {A∗ | A ∈ D} is a subgroup of Gl(D) isomorphic to D and called the adjoint group of D.

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C OROLLARY 2.2.– - O(D) is a group (the law of which is the composition of maps), - SO(D) is a subgroup of O(D) isomorphic to the group D∗ and f ∈ SO(D) ⇐⇒ f ∈ O(D) and f preserves ({X1 ; X2 ; X3 }). 2 First, let us prove the Theorem. i) ⇒ ii). Let (ξ 1 , ξ 2 ,ξ3 ) be an orthonormal  basis of D, according to i) it is evident that the three elements f (ξ 1 ), f (ξ 2 ), f (ξ 3 ) make an orthonormal system and so, an orthonormal basis od D. If X, Y, Z ∈ D and λ, μ ∈ Δ, then {f (λX + μY ) − λf (X) − μf (Y ) | f (Z)} = {λX + μY | Z} − {λf (X) | f (Z)} − {μf (Y ) | f (Z)} = {λX + μY | Z} − λ{f (X) | f (Z)} − μ{f (Y ) | f (Z)} = {λX + μY | Z} − λ{X | Z} − μ{Y | Z} = 0

With Z = ξi the previous equality proves that f (λX + μY ) − λf (X) − μf (Y ) is orthogonal to f (ξ i ) for i = 1, 2, 3, hence orthogonal to every element of D since  f (ξ 1 ), f (ξ 2 ), f (ξ 3 ) generates D by linear combinations with dual coefficients. Therefore this vector vanishes since {· | ·} is non-degenerate (Proposition 2.4). We conclude that f (λX + μY ) = λf (X) + μf (Y ). and f is Δ-linear. ii) ⇒ iii) is evident. Let us prove iii)   ⇒ iv). If (ξ 1 , ξ 2 , ξ 3 ) is an orthonormal basis of D such that f (ξ 1 ), f (ξ 2 ), f (ξ 3 ) is also an orthonormal basis, every X ∈ D may be expressed as: X = z1 ξ 1 + z2 ξ 2 + z3 ξ3 with z1 , z2 , z3 ∈ Δ. and then, since f is Δ-linear: f (X) = z1 f (ξ 1 ) + z2 f (ξ 2 ) + z3 f (ξ 3 ), so that   N (X) = z12 + z22 + z32 = N f (X) . iv) ⇒ i) is a straightforward consequence of [2.8]. At this stage the first four properties are equivalent and the proof will be complete if we prove that v) ⇒ i)

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and i) ⇒ v). v) ⇒ i) is a consequence of invariance of {· | ·} by action of D on D (Proposition 2.4). Let us prove that iii) ⇒ v). Let (ξ 1 , ξ2 ,ξ 3 ) be an orthonormal basis which is transformed into a basis f (ξ 1 ), f (ξ 2 ), f (ξ 3 ) which is also orthonormal. Let us put s = +1 when the two bases have the same orientation, s = −1 when their orientations are different so that (ξ1 , ξ2 , ξ 3 ) and sf (ξ 1 ), sf (ξ 2 ), sf (ξ 3 ) = (η 1 , η 2 , η 3 ) always have the same orientation (see section 2.1.4). Let (c; ω 1 , ω 2 , ω 3 ) and (c ; ω 1 , ω 2 , ω 3 ) (ω i = ω η ) be the orthonormal affine frames in E associated with (ξ 1 , ξ 2 , ξ 3 ) and i (η 1 , η 2 , η 3 ) respectively. There is a uniquely defined A ∈ D such that: A(c) = c ,

A(ω i ) = ω i for i = 1, 2, 3,

(the affine map is completely defined by its linear part and the first condition, the linear part A is in SO(E) because the bases (ω 1 , ω 2 , ω 3 ) and (ω 1 , ω 2 , ω 3 ) have the same orientation). Then A∗ ξ i (c ) = A(ξ(c)) = 0 = η i (c ),

ω A∗ ξ = ω i = ω η i

i

for i = 1, 2, 3.

Hence A∗ and sf are Δ-linear maps taking the same values on the member of a basis of the Δ-module D, therefore sf = A∗ . 2 Let us turn to the corollary. According to ii) if f ∈ O(D) then it is a bijective map D → D. It is evident that f ∈ O(D) ⇒ f −1 ∈ O(D), f1 and f2 ∈ O(D) ⇒ the compounded map f1 ◦ f2 ∈ O(D) we conclude that O(D) is a permutation group of D. According to v) and Proposition 2.4 if f = A∗ ∈ O(D) then f preserves the dual mixed product and f ∈ SO(D), if f = −A∗ ∈ O(D) then f reverses its algebraic sign. Indeed f ∈ SO(D) ⇐⇒ f preserves ({X1 ; X2 ; X3 }). The map A → A∗ is a monomorphism of groups from D into the group Gl(D) and, with the theorem its range is the group SO(D). Recall that an isometry of the Euclidean affine space E is an affine map A : E → E such that A ∈ O(E) (i.e. det(A) = ±1). It is a displacement when A ∈ SO(E) (i.e. det(A) = +1) and an antidisplacement when A ∈ O(E) − SO(E) (i.e. det(A) = −1). Isometries of E make a group I = I(E) (and D is a subgroup of I). There is a natural action of I on D, defined by the same formula as the action of D and such that   A∗ X : p → A X(A−1 (p) for A ∈ I. The properties of this action are the matter of exercise 2.11 below. In the following we shall meet Δ-semi-linear maps f : D → D; they verify: f (X + Y ) = f (X) + f (Y ),

f (λX) = λ f (X),

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E XERCISE 2.11.– 1) Check that, when A ∈ I, X, Y and Z ∈ D:   ω U = det(A)A ω X if U = A∗ X,   A∗ [X, Y ] = A∗ X, A∗ Y ,

[2.19]

[[A∗ X | A∗ Y ]] = det(A) [[X | Y ]],

[2.20]

{A∗ X; A∗ Y ; A∗ Z} = det(A){X; Y ; Z}

[2.21]

[2.18]

2) Prove that when A ∈ I − D the map X → A∗ X is semi-linear and that {A∗ X | A∗ Y } = {X | Y }.

[2.22]

The following properties complete Theorem 2.2: T HEOREM 2.3.– Let f be a mapping from D to D. The following properties are equivalent: i) f verifies: {f (X) | f (Y )} = {X | Y } for all X and Y in D. ii) f is Δ-semi-linear and maps every orthonormal basis of D into an orthonormal basis of D. iii) f is Δ-semi-linear and there is an orthonormal basis of D transformed by f into an orthonormal basis of D iv) f is Δ-semi-linear and verifies: N (f (X)) = N (X) for all X in D. v) f is Δ-semi-linear and there is an antidisplacement A such that either f = A∗ or f = −A∗ . In particular the preservation of the dual inner product up to a conjugation implies the semi-linearity of f . When f verifies the conditions of the Theorem and if it preserves ({X1 ; X2 ; X3 }) then f = −A∗ and if it f reverses the algebraic sign of ({X1 ; X2 ; X3 }) then f = A∗ . 2 The reasoning proving that i), ii), iii), iv) are equivalent properties are similar to those from the proof of Theorem 2.2 except that in the proof of i) ⇒ ii) we check that: {f (λX + μY ) − λf (X) − μf (Y ) | f (Z)} = 0

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and, in iii) ⇒ iv), that f (X) = z¯1 f (ξ 1 ) + z¯2 f (ξ 2 ) + z¯3 f (ξ 3 ), so that   N (X) = z¯12 + z¯22 + z¯32 = N f (X) . To prove that v) ⇒ i) we assume that f = ±A∗ with A ∈ I. Then {f (X) | f (Y )} = {±A∗ X | ±A∗ Y } = {X | Y } Let us prove that iii) ⇒ v). Let (ξ 1 , ξ2 , ξ 3 ) be the orthonormal basis which is  transformed into the also orthonormal basis f (ξ 1 ), f (ξ 2 ), f (ξ 3 ) . Let us put σ = −1 when the two bases have the same orientation, σ = +1 when  they have different orientations so that (ξ 1 , ξ2 , ξ 3 ) and σf (ξ 1 ), σf (ξ 2 ), σf (ξ 3 ) = (η 1 , η 2 , η 3 ) always have different orientations (see section 2.1.4). Let (c; ω 1 , ω 2 , ω 3 ) and (c ; ω 1 , ω 2 , ω 3 ) (ω i = ω η ) be the orthonormal affine frames in E associated with (ξ 1 , ξ 2 , ξ 3 ) and i (η 1 , η 2 , η 3 ) respectively. There is a uniquely defined A ∈ I such that: A(c) = c ,

A(ω i ) = ω i for i = 1, 2, 3,

(the linear part A is in O(E) − SO(E) because the bases (ω 1 , ω 2 , ω 3 ) and (ω 1 , ω 2 , ω 3 ) have different orientations and A is an antidisplacement). Then A∗ ξ i (c ) = A(ξ(c)) = 0 = η i (c ),

ω A∗ ξ = ω i = ω η i

i

for i = 1, 2, 3.

Hence A∗ and σf are Δ-semi-linear maps taking the same values on the member of a basis of the Δ-module D, therefore σf = A∗ . 2.2.4. Generalized Olinde Rodrigues formula In the following theorem we shall use the generalized trigonometrical functions of dual angles defined in section 2.4.2 below: if α = θ + δ (with θ and δ in R) is a dual angle then sin α = sin θ + δ cos θ,

cos α = cos θ − δ sin θ

First of all let us state some formulae L EMMA 2.2.– For all Z ∈ D (ad Z)3 + {Z | Z}ad Z = 0 in L(D) and, more generally: (ad Z)2n+1 = (−1)n {Z | Z}n ad Z (ad Z)2n

= (−1)n−1 {Z | Z}n−1 (ad Z)2

for n ≥ 0 for n ≥ 1

[2.23] [2.24]

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The first result gives the minimal polynomial of ad Z over the dual number ring; it is of degree 3. 2 With the generalized Gibbs formula [2.4] and Proposition 2.4, for all X ∈ D: ad Z 3 .X = [Z, [Z, [Z, X]]] = {Z | [Z, X]} Z − {Z | Z} [Z, X] = −{Z | Z} ad Z.X (since {Z | [Z, X]} = 0). The first formula follows and proves that [2.23] is verified for n = 0 or 1. Assume that this formula is verified for an integer n, then, since (ad Z)2 is Δ-linear:   (ad Z)2(n+1)+1 = (ad Z)2 ◦ (ad Z)2n+1 = (ad Z)2 (−1)n {Z | Z}n ad Z = (−1)n {Z | Z}n (ad Z)3 = (−1)n+1 {Z | Z}n+1 ad Z and [2.23] holds for the integer n + 1. Hence this formula is verified for all n. For n ≥ 1 [2.23] gives   (ad Z)2n = ad Z ◦ ad Z 2(n−1)+1 = ad Z ◦ (−1)n−1 {Z | Z}n−1 ad Z = (−1)n−1 {Z | Z}n−1 (ad Z)2 P ROPOSITION 2.7.– For α ∈ Δ and υ ∈ D such that {υ | υ} = 1   expD (αυ) ∗ = 1 + sin α ad υ + (1 − cos α)(ad υ)2 where ad υ is the operator X ; [υ, X] = ad υ.X. 2 With [1.105] (expD αυ)∗ = exp(ad αυ) so that it suffices to calculate the right hand side. Using Lemma 2.2 with Z = αυ and the properties of Δ-linearity {Z | Z} = α2 {υ | υ} = α2 , ad Z = α ad υ, so that 

2n+1 = (−1)n α2n+1 ad υ for n ≥ 0 ad αυ  2n = (−1)n−1 α2n (ad υ)2 for n ≥ 1 ad αυ

Therefore exp(ad α υ) = 1 +

∞ 1 (ad αυ)k k!

k=1

= 1+

∞  n=0

 (−1)n α2n+1 (2n + 1)!

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ad υ +

∞  2 (−1)n−1 2n   ad υ α (2n)! n=1

= 1 + sin α ad υ + (1 − cos α)(ad υ)2 The classical Olinde Rodrigues formula expresses a rotation R ∈ SO(E) as a function of its angle and a normalized director of its axis (see formula [1.18]). The generalization to D expresses A∗ for A ∈ D as a function of its dual angle and the normalized element of D which defines its axis in E. T HEOREM 2.4.– Let Λ be the oriented axis of the unitary screw υ in D with ω υ = u, |u| = 1 (and {υ|υ} = 1). Let A = Tr(δu) ◦ RΛ (θ) be a displacement expressed as a rotation around Λ with angle θ followed by a translation δ along Λ (the order being unimportant). Then 1) for every X in D: A∗ X = X + sin α[υ, X] + (1 − cos α)[υ, [υ, X]].

[2.25]

Or, in operator language: A∗ = 1 + sin α ad υ + (1 − cos α)(ad υ)2 ,

[2.26]

2) A = expD (αυ) with α = θ + δ.   2 If [2.26] is proved we have A∗ = expD αυ ∗ by Proposition 2.7, hence the second part follows since the map A → A∗ is one one. Let us prove the first part. Let c ∈ Λ, then D = Zc ⊕T (direct sum over R). It is sufficient to prove that the R-linear operators to the left and to the right agree on Zc and on T. But, since T = Zc and the operators are indeed Δ-linear, it is sufficient to prove only that they agree on Zc . When X ∈ Zc using the Lie algebra isomorphism Zc ↔ E, and Olinde Rodrigues formula [1.18], we have R∗ X = X + sin θ [υ, X] + (1 − cos θ)[υ, [υ, X]] (in that isomorphism, on the one hand RΛ∗ becomes R(u, θ) which may be expressed by [1.18] and, on the other hand, X ↔ ω X , υ ↔ u, [υ, X] ↔ u ∧ ω X ). Moreover, when T = Tr (δu), since T∗ X(p) = X(p − δu) = X(p) + δu ∧ ω X for all p ∈ E: T∗ X = X + δ[υ, X] Using these results, always for X ∈ Zc : A∗ X = T∗ R∗ X = X +sin θ [υ, X] + (1 − cos θ)[υ, [υ, X]]  +δ [υ, X] + sin θ [υ, [υ, X]] + (1 − cos θ)[υ, [υ, [υ, X]]] = X +sin θ [υ, X] + (1 − cos θ)[υ, [υ,  X]] +δ cos θ[υ, X] + sin θ [υ, [υ, X]]

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(because, according to [2.4], [υ, [υ, [υ, X]]] = −{υ | υ}[υ, X] = −[υ, X]). A∗ X = X + (sin θ + δ cos θ)[υ, X] + (1 − cos θ + δ sin θ)[υ, [υ, X]] = X + sin α [υ, X] + (1 − cos α)[υ, [υ, X]]. C OROLLARY 2.3.– When {υ | X} = 0, A∗ X = cos αX + sin α[υ, X]. 2 With the generalized Gibbs formula: [υ, [υ, X]] = {υ | X}υ − {υ | υ}X = −X The following lemma generalizes the classical definition of circular trigonometric fuctions and it will be completed by exercises about the definition of the dual angle between elements of D. L EMMA 2.3.– Let z1 and z2 be dual numbers such that z12 + z22 = 1. Then, there exists a “dual angles” α = θ + δ uniquely defined modulo 2πZ such that z1 = cos α,

z2 = sin α

E XERCISE 2.12.– Prove lemma 2.3, using the similar property in circular trigonometry and the formulae given at the head of section 2.2.4. E XERCISE 2.13.– Let ξ and η be two given members of D such that {ξ | ξ} = 1, {η | η} = 1, 1) Prove that there exists υ ∈ D such that {υ | υ} = 1,

{υ | ξ} = {υ | η} = 0

and that when [ξ, η] ∈ / T there is exactly two opposite normalized elements υ verifying this condition while when [ξ, η] ∈ T there is an infinity of solutions υ. 2) Check that (ξ, [υ, ξ], υ) is an orthogonal basis of D and that if η = z1 ξ + z2 [υ, ξ] is the expansion of η in this basis then z12 + z22 = 1. 3) If R(α, υ) denotes the Δ-linear operator in D defined in the right of [2.25], prove that, under the conditions of question 1, the following properties are equivalent: a) {ξ | η} = cos α and [ξ, η] = sin αυ, b) R(α, υ).ξ = η 4) Prove that there exists a uniquely defined modulo 2πZ dual angle α such that η = R(υ, α).ξ. Prove that the following properties are equivalent c) The straight lines Δξ and Δη are parallel. d) α = δ is a pure dual number.

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5) Check that if X and Y are members of D − T. then there exists a uniquely defined modulo 2πZ dual angle α = θ + δ such that: {X | Y } = N (X)N (Y ) cos α,

[X, Y ] = sin α υ

Let υ ∈ D be such that: {υ | υ} = 1,

{υ | X} = {υ | Y } = 0

and the line (a, b) be the (or a) common perpendicular to ΔX , ΔX (a ∈ ΔX , b ∈ ΔY ). Prove that θ is the angle of ω X and ω Y (in the orthogonal plane to ω υ in E) and δ is → − the distance between these axes defined as ab = δω υ . H INT.– For question 1 remark that any normalized υ ∈ D such that Δυ is perpendicular to Δξ and Δη verifies the required properties. For question 4 use question 2, question 3 and lemma 2.3.

2.3. Dual quaternions 2.3.1. Geometrical definition As is well-known, the ordinary quaternions may be defined as pairs (s, v) with s in R and v in E, the operations over them being defined by (see Chapter 1, Appendix 1) q + q = (s + s , v + v ), 





λ(s, v) = (λs, λv), λ ∈ R,







qq = (ss − v · v , sv + s v + v × v ).

[2.27] [2.28]

Endowed with operations [2.27], the set H of quaternions is a real vector space with zero = (0, 0). Endowed with the addition and multiplication [2.28] it is a field with unity e = 1 = (1, 0), and endowed with the three operation it is an associative algebra over R. It appears that the associativity of quaternion multiplication holds as a consequence of three properties of the vector product: Jacobi identity x × (y × z) + y × (z × x) + z × (x × y) = 0, skew-symmetry of mixed product x · (y × z), Gibbs formula x × (y × z) = (x · z)y − (x · y)z. Therefore this definition may be extended to the Lie algebra D over Δ which has also a natural inner product {·|·} and formally similar properties [2.3], [2.4]). Put H = Δ × D = set of pairs q = (z, X) with z ∈ Δ, X ∈ D, and: q + q = (z + z  , X + X  ), 







λ(z, X) = (λz, λX), λ ∈ Δ, 



qq = (zz − {X|X }, zX + z X + [X, X ]).

[2.29] [2.30]

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A straightforward computation, using Jacobi identity for the Lie bracket and formulae [2.3] and [2.4], the main point being the associativity, leads to the following results: T HEOREM 2.5.– Endowed with the previous addition and multiplication, H is a ring with unity (1, 0) = e (but not a field). Endowed with the first two operations it is a Δ-module and, with the three operations, an associative algebra over Δ. Endowed with addition and multiplication, H is a ring. The result would not be true if we had operated on the real Lie algebra D and the Klein form [·|·]. Thus the module and Lie algebra structure on D, over Δ and with the inner product{·|·}, are exactly those which support the geometrical concept of dual quaternions algebra. Not that H is not a field. A member (z, 0) ∈ H is a scalar dual quaternion and will be denoted by ze or simply z. A member (0, X) ∈ H is a pure dual quaternion and will be denoted [X]. If q = (z, X) ∈ H we note Sc (q) = z and Ve (q) = X the scalar and vector part of q, and we note q = (z, −X). For example next formula holds ¯1, ¯2q q 1 q2 = q

Sc (q1 q2 ) = Sc (q2 q1 ),

Sc ([X][Y ]) = −{X|Y },

Ve ([X][Y ]) = [X, Y ].

[2.31] [2.32]

Moreover a member of H which commutes with all other members of H is scalar. 2.3.2. Norm and invertibility in H For q in H define: N (q) = qq = qq = (z 2 + {X|X})e. Then, N (q) is a scalar dual quaternion with positive real part. Easy computations lead to the fundamental properties N (qq ) = N (q q) = N (q)N (q ),

N (q) = N (q).

[2.33]

The dual number N (q) is invertible if and only if it has a non vanishing real part, therefore q ∈ H is invertible if and only if (N (q)) > 0 and q−1 = N (q)−1 q.

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2.3.3. Dual quaternions and representation of D in D The following theorems provide a representation of spatial displacements as operations on D by means of dual quaternions as defined here: T HEOREM 2.6.– If q is in H and N (q) = 1 there exists A ∈ D such: [A∗ X] = q[X]q for every X ∈ D. Note A∗ = D(q) the relation defined in this theorem. T HEOREM 2.7.– For every A in D, there exist two and only two normalized dual quaternions q and −q which are mapped onto A∗ (i.e A∗ = D(q) = D(−q)). T HEOREM 2.8.– If A∗ = D(q), A∗ = D(q ) then: (A ◦ A )∗ = D(qq ). Thus, the product of displacements is related to the product of dual quaternions. 2 From formula [2.31] we deduce Sc (q[X]q) = Sc (qq[X]) = Sc ([X]) = 0, therefore q[X]q = [Y ] is a pure dual quaternion with a well defined Y in D. It is easy to show that the relation between X and Y is a Δ-linear map D(q), so q[X]q = [D(q)X]. Moreover, if D(q) maps X, X  , X  into Y , Y  , Y  respectively, then {Y |Y  } = {X|X  },

{Y |[Y  , Y  ]} = {X|[X  , X  ]}.

The proofs of both formula are similar. For instance: {Y |Y  } = −Sc ([Y ][Y  ]) = −Sc (q[X]qq[X  ]q) = −Sc (qq[X][X  ]) = −Sc ([X][X  ]} = {X|X  }. Thus D(q) preserves the forms {·|·} and {.|[., .]} and Theorem 2.6 follows from Theorem 2.2 equivalence i) ⇔ v). In Theorem 2.7 the existence part is a consequence of two properties. If q = (z, ζ) ∈ H and N (q) = 1 then an easy computation using formula [2.4] and z 2 + {ζ|ζ} = 1 shows that (notation of Theorem 2.4): D(q)(X) = X + 2z[ζ, X] + 2[ζ, [ζ, X]] with q = (z, ζ).

[2.34]

If α = θ + δ ∈ Δ, υ ∈ D, {υ|υ} = 1, D(q) = (T r(δi) ◦ RΛ (θ))∗ , with q = (cos α/2, sin α/2 υ). then the second follows from the first one and Theorem 2.4.

[2.35]

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If D(q) = D(q ) for two normalized dual quaternions, then qq [X] = [X]qq for all X in D. Thus qq commute with all quaternion [X] therefore with all members of H, and it is a scalar dual quaternion (z, 0). Being normalized z = ±1. Thus qq = ±e and q = ±q. The proof of Theorem 2.7 is complete. Theorem 2.8 is obvious. This theorem, with property [2.35], leads to a classical formula giving the geometrical elements of a product of displacements (see [HIL 84]). 2.4. Differential calculus in Δ-modules 2.4.1. Δ-differentiable maps The following form of the definition of differentiable maps of dual variables has been used in reference [HAM 93] by A. Hamlili and does not assume that such a function can be expanded into a series (assuming a priori an expansion into a series is not correct from the rigorous mathematical standpoint; in a consistent theory, if it exists, such an expansion must follow from a theorem). This definition solves completely the problems associated with the differentiability of a function of dual variables and allows a deduction from a more basic principle of the formula of Definition 1 in Rico and Duffy [RIC 93]. Extensions of the functions of real variables to variables belonging to rings were defined by A. Burov in [BUR 88] and used for other mechanical purposes (the search for integrals of Hamiltonian systems). In the sequel, we have to assume that norms - in the common meaning - are defined  or F.  For example, we may put on extended Δ-modules E z = x + y

 x, y ∈ E. for x + y = z ∈ E,

 we take a norm on the real vector space E × E (as In other words as a norm on E it is well known, as far as we consider finite dimensional vector spaces, all the norms are equivalent and the following results are independent of the choice of the norms).  to F  is said to be Δ of E D EFINITION 2.3.– A map f from an open subset U   such that  differentiable at z ∈ U if there exists a Δ-linear map A from E to F  (with z + δz ∈ U  ). [2.36] f (z + δz) = f (z) + A(δz) + o(δz) for all δz ∈ F where o(δz) denotes a function such that

lim

δz→0, δz =0

o(δz) = 0. δz

Relation [2.36] is similar to the definition of an ordinary differentiable map except that we assume that the part δz → A(δz), the differential of f at z, is not only a R-linear map but also a Δ-linear map (a strong but very natural limitation). A Δdifferentiable map is also an ordinary differentiable map within the meaning of real analysis (of course, the converse is untrue).

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 be  → F P ROPOSITION 2.8.– A necessary and sufficient condition for a map f : U  is that f is differentiable at (x, y) according to Δ-differentiable at z = (x, y) ∈ U real analysis and that the following relations hold: ∂g ∂h (x, y) = (x, y), ∂x ∂y

∂g (x, y) = 0, ∂y

[2.37]

for the real and dual part of f defined by f (z) = g(x, y) + h(x, y) if z = (x, y) ≡ x + y. Properties [2.37] are analogous to the Cauchy-Riemann equations encountered in the theory of holomorphic functions. ∂g ∂g ∂h (x, y), (x, y), (x, y), are linear operators from E to ∂x ∂y ∂y F which may be described in coordinates by Jacobian matrices with respect to the coordinates of x or y: The quantities



 ∂gi (x, y) , ∂xj



 ∂gi (x, y) , ∂yj



 ∂hi (x, y) . ∂yj

2 If f is a differentiable map according to real analysis, its differential map at z is the R-linear map defined by ∂f ∂f (x, y).δx + (x, y).δy ∂x ∂y     ∂g ∂g ∂h ∂h = (x, y).δx + (x, y).δy +  (x, y).δx + (x, y).δy . ∂x ∂y ∂x ∂y

A(δz) =

As in the proof of [2.15] we see that a necessary and sufficient condition for this map be Δ-differentiable is that the conditions [2.37] hold. 2.4.2. Extensions of ordinary differentiable maps into Δ-differentiable maps  are a real vector space and the extended module, if U is a subset of When E and E  the subset U  = U × E of of E × E, in other words E we shall denote by U  ⇐⇒ Re(z) = x ∈ U. z = x + y ∈ U   is open in E. If U is open in E then U

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115

 and F  the extended modules T HEOREM 2.9.– Let E and F be real vector spaces, E over Δ. Let f be a twice differentiable map from an open subset U of E into F (or   such that:  to F F).Then there exists a uniquely defined extended map fˆ from U i) fˆ is a Δ-differentiable map, ii) fˆ(x) = f (x) if z = x is real and belongs to U .  , then: If z = x + y ∈ U ∂f fˆ(z) = f (x) +  (x) · y. ∂x

[2.38]

Note that, in the right hand side of [2.38], f may be a dual-valued function. 2 Write the given function f (x) = φ(x) + ψ(x) with φ(x), ψ(x) ∈ F for x ∈ U . According to Proposition 2.8, for fˆ(x + y) = g(x, y) + h(x, y) be an extended map with the required properties, it is necessary and sufficient that: ⎧ ∂g ⎪ ⎪ (x, y) = 0, ⎪ ⎨ ∂y

g(x, 0) = φ(x),

⎪ ⎪ ∂h ∂g ⎪ ⎩ (x, y) = (x, y), h(x, 0) = ψ(x). ∂y ∂x  , (this The conditions in the first line imply g(x, y) = φ(x) for all x + y ∈ U  ). conclusion can be drawn since the straight line from x to x + y always lies in U Now, the conditions in the second line imply: ∂h ∂φ (x, y) = (x), ∂y ∂x hence, for all fixed x ∈ U , the differential of the map y → h(x, y) −

∂φ (x).y ∂x

vanishes. This map is constant and equal to ψ(x) giving the value of h(x, y). Finally,   ∂f ∂φ ˆ (x).y = f (x) +  (x).y . f (x + y) = φ(x) +  ψ(x) + ∂x ∂x

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conversely, if f is twice differentiable, this formula defines a differentiable map within the meaning of real analysis and this map is Δ-differentiable (condition 2.37). If we take a basis in E and use coordinates x1 , . . . , xn in E and the corresponding  then formula [2.38] becomes: coordinates z1 = x1 + y1 , . . . , zn = xn + yn in E, fˆ(x + y) = f (x) + 

n

yi

i=1

∂f (x). ∂xi

If we also take a basis in F, the coordinates of fˆ(x + y) relative to the  are corresponding basis of F fˆj (x + y) = fj (x) + 

n i=1

yi

∂fj (x), ∂xi

j = 1, . . . , p.

E XAMPLE 2.1.– The dualization of R as a real vector space gives Δ and if U is an  = U ×R. Theorem 2.9 leads to the extensions of the classical open subset of R, then U functions of a real variable: – Let f (x) = exp(kx) (k ∈ R) defined on R then f: z = x + y → exp(kx)(1 + ky) It is readily verified that exp(z1 + z2 ) = exp z1 × exp z2 for z1 and z2 in Δ. – Let f (x) = ln x defined on U =]0, +∞[ then f is defined on ]0, +∞[ ×(R) by y f: z = x + y → ln x +  . x – Let f (x) =

√

x defined on U =]0, +∞[ then f is defined on ]0, +∞[ ×(R) by

f: z = x + y →

√ y x+ √ . 2 x

For convenience we define the square root of the dual number 0 as order that N (X) to be always defined for X ∈ D).

√ 0 = 0 (in

– By the same way we obtain the extensions of trigonometric functions: sin z = sin x + y cos x,

cos z = cos x − y sin x,

tan z = tan x + y(1 + tan2 x).

Dual Numbers and “Dual Vectors” in Kinematics

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– Let x → P (x) be a polynomial function with real coefficients then Pˆ (z) = P (x) + yP  (x). and we see that the zeros of P in Δ are its real zeros plus, perhaps, the infinity of dual numbers of the form xo + y where xo is a real zero of P and P  and y ∈ R.

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3 The “Transference Principle”

3.1. On the meaning of a general algebraic transference principle Two frameworks were defined in Chapters 1 and 2: 1) The real vector spaces E, endowed with the vector product ∧ and the Euclidean scalar product, and its isomorphic Lie subalgebra Zo of D endowed with the Lie bracket and the Killing form (·|·). The members of E are ordinary vectors, those of Zo are line vectors through the point o. According to this framework the significant groups are the isomorphic groups SO(E) and Rot(o)∗ (the group of the R∗ with R ∈ Rot(o)).  and D endowed with their additional structures of Lie 2) The Δ-modules E  and the dual inner product {· | ·} built algebras over Δ, the dual inner product in E  are dual vectors with the Killing form and the Klein form in D. The members of E and those of D are screws. According to this framework the significant groups are the  and D∗ (the group of the A∗ with A ∈ D). isomorphic groups SO(E) A form of the transference principle would say that every statement in the first framework can be translated into an equally valid statement in the second framework (see for example [ROO 78]). However it is necessary distinguish carefully a mere “dualization principle”, translating true or not propositions into similar propositions in the second framework (as was for example the Kotelnikov-Dimentberg principle in [DIM 65]), and a principle which would translate theorems into equally valid similar theorems. Understood within the latter meaning, unfortunately, the principle does not hold; simple examples illustrating its failure follow. 1) The law on real numbers and line vectors through o: if λ ∈ R, X ∈ Zo and λX = 0 then necessarily λ = 0 or X = 0 is true, but the statement on dual number and screws “if λ ∈ Δ and X ∈ D and λX = 0 then necessarily λ = 0 or X = 0” is untrue.

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2) The statement: if the elements X1 , . . . , Xn of Zo are linearly dependent over R then one of them is a linear combination of the other ones is true, but the corresponding statement: “if the elements X1 , . . . , Xn of D are linearly dependent over Δ then one of them is a linear combination of the other ones” is untrue. 3) The law concerning lines through o: if X = 0 and Y are in Zo then [X, Y ] = 0 ⇐⇒ there exists λ ∈ R such that Y = λX, is equivalent to a well-known property of real vectors but the translated property concerning any lines or screws in space: if X = 0 and Y are in D then [X, Y ] = 0 ⇐⇒ there exists λ ∈ Δ such that Y = λX, is not true. In fact if we take X ∈ T, X = 0 then [X, Y ] = 0 for all Y in T (and not necessarily of the form λX). 4) The law: if X is in Zo then (X | X) = 0 is equivalent to X = 0 is true but the translated property: if X is in D then {X | X} = 0 is equivalent to X = 0 is untrue (the right property should be {X | X} = 0 is equivalent to X is in T). 5) The theorem on line vectors through o: If U and V are in Zo and U = 0 then, a necessary and sufficient condition for the equation [U, X] = V be solvable with respect to X is that (U | V ) = 0 is true. Its translation on screws: If U and V are in D and U = 0 then, a necessary and sufficient condition for the equation [U, X] = V be solvable with respect to X is that {U | V } = 0 is untrue. An analysis of this “principle of transference” was presented with more details in reference [CHE 96].  3.2. Isomorphy between the adjoint group D∗ and SO(E) The adjoint group D∗ of D is the group of linear operators Ad A = A∗ acting on the Lie algebra D of D. It was already proved that it is isomorphic with D itself and that D∗ and SO(D) are isomorphic subgroups of Gl(D) (section 2.2.3 and Corollary 2.2-1), the complete result is:  SO(D) and D∗ are isomorphic. T HEOREM 3.1.– The groups SO(E), 2 Using Proposition 2.6 and its Corollary it is readily proved that the mappings w → Jo−1 ◦ w ◦ Jo = ϕ,

ϕ → Jo ◦ ϕ ◦ Jo−1

 and the reciprocal are an isomorphism of the group SO(D) onto the group SO(E) isomorphism. Since isomorphy with D∗ was already proved the three groups are isomorphic.

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The displacement group plays a part in the computations of kinematics and dynamics only through its adjoint group, so it is useful to mechanics to have at  allowing one’s disposal an expression of the isomorphism between D∗ and SO(E) to transform calculations with the displacement group into calculations with Δ-linear operators and dual vectors or with dual matrix. In matrix form the relation between orthogonal dual operators and displacements is Veldkamp’s theorem (see [VEL 76]); this is the matter of the following exercise (in Veldkamp’s original form the operators ˆ2 , e ˆ3 ) of are expressed by 3 × 3 matrix of dual numbers with respect to a basis (ˆ e1 , e  E associated with an orthonormal basis (e1 , e2 , e3 ) of E.) E XERCISE 3.1.– 1) Check that when o is a fixed point of E, R ∈ Rot(o) is a rotation  is the extended operator of R ∈ ˆ ∈ SO(E) about o, R ∈ SO(E) is its linear part, R SO(E) and T = Tr (u) is a translation:     ˆ o (X) and Jo T∗ X = (1 + ˆ for all X ∈ D: Jo R∗ X = RJ u∼ )Jo (X). ˆ is u as dual vectors and u ˆ ∼ the Δ-linear operator z → u ˆ ∧ z. where u 2) Check that if a displacement is splitted into a rotation and a translation as D =  ˆ ˆ ∈ SO(E) R◦Tr (u) ≡ Tr (v)◦R with R ∈ Rot(o), then R.(1+ˆ u∼ ) = (1+ˆ v∼ ).R with v = R(u) and, if ϕ is this orthogonal Δ-linear operator, then for all X ∈ D:   Jo (D∗ X) = ϕ Jo (X) .  → E  is a Δ-linear map which is skew-symmetric (for 3) Prove that, if f : E  such that for all z ∈ E:  the extended dot product) then, there is a unique a ∈ E f (z) = a ∧ z (with the extended vector product).  there is a uniquely defined 4) Prove that, conversely, when ϕ ∈ SO(E) displacement D such that:     Jo D∗ X = ϕ Jo X for all X ∈ D. and that the relation ϕ ↔ D is an isomorphism of groups (depending of the choice of the origin o).  H INT.– Write ϕ = A + B where A and B are in L(E) and prove that, if ϕ ∈ SO(E) then A−1 ◦ B and B ◦ A−1 are skew-symmetric operators and use question 3.) 3.3. Regular maps Now we turn to properties of transference involving differential calculus and group structures. Let us remind some properties of kinematics with the notation of formula [1.14], section 1.2.2. Let t → R(t) be a derivable map from R to SO(E)(E) (in

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kinematics, a motion of a body performing rotations about a fixed point), then for all t, the angular velocity ω(t) is such that d ∼ R(t) = ω (t) ◦ R(t) dt Let t → D(t) be a derivable map from R to D (in kinematics, a general motion of a rigid body), then for all t, according to Theorem 1.13, there exists V (t) ∈ D such that d D(t) = V (t) ◦ D(t) dt The vector field V (t) is the velocity vector field or “the rate of twist” at time t. → − Moreover, the angular velocity ω(t) such that V (t)(a) = V (t)(b) + ω(t) ∧ ab for a, b in E could also be defined as above from the linear part R(t) = D(t) of D(t). As we have seen in Chap. 1, Proposition 1.8, these properties extend to differentiable maps of several real variables: let U be an open subset of Rn (n non zero integer) and let θ = (θ1 , . . . , θn ) ∈ U → R(θ) ∈ SO(E),

[3.1]

θ = (θ1 , . . . , θn ) ∈ U → D(θ) ∈ D,

[3.2]

be differentiable maps, then, for all θ in U , there exist vectors ω 1 (θ), . . . , ω n (θ) such that: ∂R ∼ (θ) = ω k (θ) ◦ R(θ), k = 1, . . . , n, ∂θk

[3.3]

and members W1 (θ), . . . , Wn (θ) of D such that: ∂D (θ) = Wk (θ) ◦ D(θ), k = 1, . . . , n. ∂θk

[3.4]

(In kinematics of motions depending of several parameters, ω k (θ) is the “partial angular velocity” and Wk (θ) is the “partial rate of twist” corresponding to the variable θk ). D EFINITION 3.1 (Regular maps).– For a map of the form [3.1], the point θ ∈ U is said to be regular if the vectors ω 1 (θ), . . . , ω n (θ) in [3.3] are linearly independent (then, necessarily 1 ≤ n ≤ 3). For a map of the form [3.2], the point θ ∈ U is said to be regular if the elements W1 (θ), . . . , Wn (θ) of D in [3.4] are linearly independent (then, necessarily 1 ≤ n ≤ 6).

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123

A point θ ∈ U is said to be singular if it is not regular (the vectors ω 1 (θ), . . . , ω n (θ) or W1 (θ), . . . , Wn (θ) are linearly dependent). A map of the form [3.1] or [3.2] is regular if all the points belonging to its domain U are regular. This definition of regular and singular points is of great significance in kinematics (see Chap. 5). In particular it agrees with the standard terminology used in the following for the shape functions of kinematical chains and when the map [3.2] defines the position of the end-effector on an n-degrees-of-freedom manipulator. It is also of significant with respect to differential geometry of the groups SO(E) or D according to the following theoretical results (the open subsets in SO(E) or D are defined according to their natural topology of metric spaces): T HEOREM 3.2.– The injective regular maps from open subsets of R3 onto open subsets of SO(E) are the charts of a manifold structure on SO(E). The injective regular maps from open subsets of R6 onto open subsets of D are the charts of a manifold structure on D 1. On some neighborhood of each point of its domain, every regular map induces an injective regular map (i.e. a chart of the manifold SO(E) or D if n = 3 or n = 6 according to the context). In practice, this theorem provides us with a simple general criterion, really suited to the framework of kinematics, for proving that a local “parameter representation” is a chart of the natural Lie group structure of SO(E) or D. The third part of the theorem shows that when one uses parametric representations of these groups, the most important property is the regularity implying that, on small enough subsets, the relation between the parameters and the element of the group ant its reciprocal one-one and differentiable. A complete proof of the theorem needs a careful analysis and is beyond the scope of the present book. 3.4. Extensions of the regular maps from U to SO(E) This section is devoted to the proof of Theorem 3.3 below, a quite general form of a “principle of transference”, including algebraic and differential aspects. J.Rooney in [ROO 78] draws up a parallel between a list of representations of rotations by real parameters and a list of representations of general spatial displacements by dual 1 In the standard terminology of differential geometry, the word “chart” means a map from an open subset of the manifold onto an open subset of a space Rn . In the theorem, we use the same word for the reciprocal maps, nevertheless, no error can result from such a slight change in the terminology which suits our purposes better.

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paameters. All of them are in accordance with this result, however the present theorem is general. We start with a parameter representation of orthogonal operators (members of the special orthogonal group SO(E) acting on vectors of E) and it is considered as a parameter representation of rotations about some point o and acting on points of E. The aim will be to extend this representation of SO(E) by means dual parameters whereas the rotations about o are extended into displacements. To be accurate, letters in boldface will denote members of SO(E) in the following and let o be a fixed point in E. We note jo the map SO(E) → Rot(o) such that A = jo (A) is the uniquely defined rotation about o verifying Linear part of A = A; jo is the reciprocal of the group isomorphism Rot(o) → SO(E)2. T HEOREM 3.3.– Let E be an Euclidean vector space of dimension n, E be an affine space over E and let o be any origin in E. Let R be a differentiable map from an open subset U of Rn into  L(E) such that R(θ) ∈ SO(E) for all θ = (θ1 , . . . , θn ) in U . Then R = jo ◦ R : U → Rot(o) extends into a differentiable map D from  = U × Rn to D (considered as a subset of Δn ) with the following the subset U properties:  , the linear part of D(ζ) is R(θ). More specifically: a) For ζ = θ + ξ ∈ U D(ζ) = Tr (v(θ, ξ)) ◦ R(θ) with v(θ, ξ) =

n 

ξk ω k (θ),

k=1

ω k (θ) defined as in [3.3].  , D(ζ) = R(θ) ∈ Rot(o) (roughly speaking, when b) For ζ = θ + 0 “real” in U ζ = θ is real, D(ζ) reduces to the rotation R(θ) about o), c) The set of the regular points (respectively singular points) of the extended map D is the set of ζ = θ + ξ such that θ is a regular point (respectively singular points) for R. In particular, if the map R is regular, then the extended map D is also regular.  = U × Rn may be considered as a subset of Δn if we do not make Moreover U   the difference between the elements of the form (θ1 , . . . , θn ), (ξ1 , . . . , ξn ) with (θ1 , . . . , θn ) ∈ U and the elements of the form (θ1 + ξ1 , . . . , θn + ξn ), sequence of dual numbers, making a subset of Δn . For n = 3 the theorem leads to the:

 → −−−−→ om for m ∈ E. 2 More specifically, A(m) is defined by oA(m) = A −

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125

C OROLLARY 3.1.– Every regular parameter representation of an open subset of the rotation group SO(E) by three real parameters (a chart of the manifold SO(E)) gives rises to a parametric representation by three dual parameters of an open subset of the displacement group D (a chart of the manifold D by six real parameters). So, all the charts of a rotation group may be extended into charts of the displacement group but the extended charts form a particular class among the charts of the displacement group. The proof of Theorem 3.3 will be given in two lemmas: L EMMA 3.1.– Let R be a differentiable map from an open subset U of Rn into L(E) such that R(θ) ∈ SO(E) for all θ = (θ1 , . . . , θn ) ∈ U,  LΔ (E)  satisfies :U  → L(E) then the extended map R  for all ζ = θ + ξ ∈ U  . R(ζ) ∈ SO(E) 2 By Theorem 2.9 applied to the map θ → R(θ) ∈ L = L(E), the extended map is defined by  + ξ) = R(θ) +  ∂R (θ)(ξ) R(θ ∂θ

 (∈ L),

 the Δ-linear and the right hand side may be thought off as a member of LΔ (E), operator such that ∂R z = x + y → R(θ).z +  (θ)(ξ). ∂θ   ∂R (θ)(ξ).x . z = R(θ).x +  R(θ).y + ∂θ  Now, we have to prove that, for the extended scalar product on E:   + ξ).z2 = z1 · z2 for all z1 , z2 ∈ E.  + ξ).z1 · R(θ R(θ

[3.5]

The left hand side of [3.5] is equal to  R(θ).z1 · R(θ).z2 + 

 ∂R ∂R (θ)(ξ).x1 · R(θ).x2 + R(θ).x1 · (θ)(ξ).x2 . [3.6] ∂θ ∂θ

Since, by the assumptions of the theorem, the first term writes R(θ).z1 ·R(θ).z2 = z1 · z2 , and since the term inside the braces is the value at ξ of the differential of the

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constant map θ → R(θ).x1 · R(θ).x2 = x1 · x2 , it vanishes. Therefore formula [3.5] is proved and Lemma 3.1 too.  At this stage the mapping ζ → jo ◦ R(ζ), takes its values in the group D. L EMMA 3.2.– Under the assumptions of Lemma 3.1, a necessary and sufficient   be a regular point for the extended map jo ◦ R condition for a point ζ = θ + ξ ∈ U is that θ be a regular point for R. 2 By the assumptions ∂R (θ) = ω k (θ) × R(θ) for θ ∈ U, k = 1, . . . , n, ∂θk and ω 1 (θ), . . . , ω n (θ), are linearly independent vectors in E. Therefore,  + ξ) = R(θ) + Λ(θ, ξ) × R(θ) with Λ(θ, ξ) = R(θ

n 

ξk ω k (θ).

k=1

 + ξ) is a regular map from a subset of Now, we show that the map (θ, ξ) → R(θ  R to SO(E). By some simple calculations: 2n

 ∂R  + ξ) (θ + ξ) = ω k (θ) × R(θ ∂θk   ∂Λ  + ξ). + (θ, ξ) + ω k (θ) × Λ(θ, ξ) × R(θ ∂θk  ∂R  + ξ). (θ + ξ) = ω k (θ) × R(θ ∂ξk Finally,  ∂R  + ξ), (θ + ξ) = vk (θ, ξ) × R(θ ∂θk

 ∂R  + ξ), (θ + ξ) = wk (θ, ξ) × R(θ ∂ξk

putting  vk (θ, ξ) = ω k (θ) + 

 ∂Λ (θ, ξ) + ω k (θ) × Λ(θ, ξ) , ∂θk

wk (θ, ξ) = ω k (θ).

As it is readily proved, these dual vectors are linearly independent over R if and only if the vectors ω k (θ) are linearly independent over R.

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127

 to D For proving the theorem, we translate the result of Lemma 3.2 from SO(E)  using the isomorphism jo . Define D(θ, ξ) = jo ◦ R(θ + ξ) and Wk (θ, ξ) for θ ∈ U by: Jo (Wk (θ, ξ)) = vk (θ, ξ),

k = 1, . . . , n,

then, as it is readily checked: ∂D (θ, ξ) = Wk (θ, ξ) ◦ D(θ, ξ), ∂θk

∂D (θ, ξ) = (ΩWk (θ, ξ)) ◦ D(θ, ξ). ∂ξk

Now, putting Wk (θ, ξ) = ΩWk−n (θ, ξ) for k = n + 1, . . . , 2n, the family {Wk (θ, ξ) | k = 1, . . . , 2n} is linearly independent in D for all (θ, ξ) in U × Rn and the proof of the theorem is over. The same method should apply to proving a little more general result which will be used in the following:  = U × X. T HEOREM 3.4.– Let X be a real vector space, U be an open set of X and U Let R be a differentiable map from U into L(E) such that R(x) ∈ SO(E) for all  LΔ (E)  be the extended map, then: :U  → L(E) x ∈ U , and let R  for all z = x + y ∈ U  , 1) R(z) ∈ SO(E) 2) For every choice of an origin o in E, R = jo ◦ R extends into a differentiable  to D, with the following properties, map D : z = x + y → D(z) from U a) The linear part of D(z) is R(x), b) If z = x + 0 is real, then D(z) ∈ Rot(o), c) The set of the regular points [singular points] of the extended map D is the set of z = x + y such that x is a regular point [singular point] for R. In particular, if the map R is regular, then the extended map D is also regular.  : D(z) = Tr (v(x, y)) ◦ R(x), where R(x) ∈ Rot(o) Moreover, if z = x + y ∈ U and v(x, y) ∈ E are defined by linear part of R(x) = R(x),

∼

v (x, y) ◦ R(x)−1 =



∂R (x).y ◦ R(x)−1 . ∂x

The present result agrees with the previous ones. If we use coordinates in X: n

n

k=1

k=1

  ∂R ∂R ∼ yk (x) = yk ω k (x) ◦ R(x). (x).y = ∂x ∂xk

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R EMARK 3.1.– If U = X = Zo and R is the exponential map exp : Zo → Rot(o)  = D and by dualization we get the general exponential of a rotation group, then U map exp : D → D of the displacement group. This result could also be deduced from formulas given in [CHE 91].

4 Kinematics of a Rigid Body and Rigid Body Systems

4.1. Introduction This chapter and the following two are devoted to kinematics. It starts with the kinematics of a single rigid body, which is a natural continuation of the chapter on Lie groups, especially on the Euclidean group D. Indeed the set S of positions of a rigid body, also called the configuration space of the body, is a principal homogeneous space of D, what is meaning that there is a left-action of D on S with the properties explained below in section 4.3 [4.1]. Some mathematical symmetry properties of the calculations in a Lie group disappear in S, what demonstrates the differences between Lagrangian and Eulerian pictures of mechanics. Fortunately the differential calculus is easily extended from D to S and it leads to nice coordinate free formulas for the velocities and acceleration of a rigid body as well in Eulerian picture as in Lagrangian picture. The relationships between the two pictures is now explicit and clear: it consists in an adjoint transformation in the Lie group D (for more details see article [CHE 00]). The first problem raised by mechanics of systems, is to know the mathematical structure of their configuration space. The group D again acts on this space in a natural manner but the fundamental property of the configuration space of a rigid body to be an homogeneous space is no longer verified because this natural action is not transitive and new tools are necessary1. This structure strongly depends on the nature of the joints between the bodies of the system and of the general organization of the system. In the following we only consider joints which are completely defined by Lie subgroups of D (kinematic pairs) producing holonomic constraints. This assumption is in agreement with the calculations in Lie group formalism, it preserves their efficiency, it permits the description of a very large class of systems and the lower 1 They are exposed in Chevallier [CHE 07] and [CHE 11].

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pairs according to the terminology of Reuleaux [REU 63] enters into this model. With the mathematical tools exposed in this section, the reader will be able to tackle the dynamics of rigid body which will be expanded in the framework of Lie groups and homogeneous space in Chapters 7 and 8 after the chapters on kinematics. Section 4.6 is devoted to the mathematical features of kinematics of systems of rigid bodies, mainly of kinematics of chains of bodies (links connected by joints). Usually a graph is attached to such a system; graph theory was introduced in multibody mechanics by Wittenburg (see [WIT 77]). The necessary elements of this theory are recalled in section 4.6.1, they will be applied to kinematics of chains in section 4.7 and again to dynamics of rigid body systems in Chapter 7. section 4.6.3 investigates the description of relative motions of two bodies connected together by a kinematic pair in the language of Lie groups. In particular the calculations of accelerations which will be used used in dynamics are developed2. The concept of graph of a mechanical system leads to distinguish between two species of chains : closed and open chains corresponding to cycles and open paths of the graph. Both kinds of chains will be investigated in the following chapters; the most difficult case, closed chains which deals with the well-known mechanisms theory, needs a particular attention and it will be postponed to the last two sections of the chapter. Chapter 5 is devoted to a systematic analysis of the kinematic of open chains and two main issues are raised: singularities and inverse kinematics. Kinematics of serial robots is a natural range for applications of these issues. We take the opportunity to bring more Lie group language into the ordinary notions of kinematics and to point out the role of the “shape function” or “work function” associated to a kinematic chain and some general results which may be deduced from its properties. Chapter 6 deals with closed chains the graphs of which are cycles. From the kinematic point of view, the matter is no more than the well-known mechanism theory. Now the central role is played by the “closure function” of the mechanism. For a part the results of the previous sections apply to this function and, we rely on them to present a complete classification of the singularities of mechanisms; this classification is fundamentally local “in the small”. Afterwards, local results “in the large” for four or five bodies mechanisms illustrate other aspects of the efficiency of calculations in Lie groups.

2 Note that it is conceivable to extend the model of chains including a finite number of bodies which is presented here - into a continuous model including an infinite number of bodies in order to obtain a one-dimension Cosserat medium and a theory of beams. See [BOY 12].

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4.2. Kinematics of a rigid body At first sight, when abstract geometry is known and assumed to be applicable to physical space, a rigid body appears as an object retaining all its geometrical properties in all circumstances. In fact, there is a strong connection between origin of Euclidean geometry and existence of rigid object in nature. According to this idea, Henri Poincaré considers that the root of our geometry lies in the properties of the Euclidean displacement group and that, in a sense, this group and geometry after are derived from handling of bodies considered to be “rigid”. In other words, for the French mathematician, the concept of rigid body is prior to geometry (see [POI 02]). For Newton in [NEW 11] and Euler in [EUL 60], for the needs of kinematics (and dynamics), rigid body were described as aggregates of interacting particles moving in Euclidean space and this way of exposition has been followed in all the classical treatises of mechanics. But such a model, with the picture of internal forces acting between particles and preserving geometrical properties in all circumstances, is far from reality and, after all, it is not necessary. In a more recent approach Vallée and all [VAL 99], consider rigidity as an additional constraint applied to a system of particles moving according to the affine group and they derive the reaction due to this constraint. Kinematics of affinely deformable bodies in the framework of Lie group theory was also expanded by Bourov and Chevallier [BOU 08]. In the present chapter we show that only the mathematical properties of the displacement group are actually involved in the kinematics of rigid body and rigid body systems. It is only in section 4.4, when we shall aim at showing the link with the usual exposition of mechanics, that we shall refer to a model of body as a set of particles; at the end, we shall notice that the mathematical structure of kinematics (and further of dynamics) will retain nothing from this model. 4.3. The position space of a rigid body Let S be the space depicting in a mathematical language the positions of a rigid body seen by an observer. To be clear, any observer uses some devices to locate the positions of objects in space, it can perform some measures and uses some set of parameters to locate the body (as coordinates of the position of a point of the body and three angles defining its orientation in space). After all, the process amounts to measure the displacements of the observed body with respect to a reference body fixed in the “laboratory”. But there exists an infinity of ways to do this and the points of S describe the positions leaving aside the choice of the parameters.

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The main property of S is the following: there is a left action of the Euclidean displacement group on S and this action is free and transitive. In other words, there is an operation D × S → S : (g, s) → g.s such that, for all s ∈ S and for all g, h ∈ D: ⎧ ⎨ e.s = s, g.(h.s) = (gh).s, [4.1] ⎩ for all fixed r ∈ S the map ν r : g → g.r from D to S is one-one and onto. The first two properties are those of an action of the group D, the third means that this action is free and transitive. In concrete terms g.s describes the result of the displacement g acting on the position s and the three statements [4.1] mean that S is a principal homogeneous space of D and reflect obvious properties of a rigid body. Namely the first property says that the action of the identity e changes nothing regarding the position, the second one means that the action of g after the action of h on the position s is the same as the action on s of the product gh in the group D, the last one says that on the one hand g.s = s only when g = e, on the other hand if r is some fixed position any other position may be obtained from r by action of a (uniquely defined) displacement3. Kinematics and dynamics need differential calculus. For a mathematician, assuming that the differential calculus in the Lie group D is known, a natural way to develop kinematics and dynamics within this frame would be to endow S with a manifold structure derived from that of D thanks to the third property. However we want to avoid such technique and we shall limit ourself to state the following rules which amount to those possibly derived from a general theory. The statements will be given for C 1 maps taking their values in a vector space but similar rules are allowed when the vector space V is replaced by an affine space. D EFINITION 4.1 (Regulatity of maps defined or valued in S).– – Let U be an open subset of Rn (n ≥ 1). A map ϕ : U → S is said to be C 1 (i.e. continuously differentiable) if for one fixed r ∈ S the map Φr : U → D such that ϕ(x) = Φr (x).r is C 1 as a map from U to D. – Let V be a vector space (over R). A map f : S → V is said to be C 1 if for one fixed r ∈ S the map Fr : D → V such that Fr (g) = f (g.r) is C 1 as a map from D to V. The statements of definitions of all other forms of regularity (continuity, differentiability at some order k ≥ 1 of ϕ or f at one point of their domain and so on) follow the same way. In spite of their appearance, those properties are independent of 3 For practical applications, property “g.s = s only when g = e” is incompatible with the picture of non realistic bodies whose positions are schematically described by points or lines. This is in harmony with the will to present a direct approach to kinematics and dynamics of rigid body.

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the choice of r in S (according to exercise 4.2 the terms “for one fixed r ∈ S” might be replaced by “for all fixed r ∈ S”). In the framework of [4.1], a motion of a rigid body is defined by a map t → s(t) from R (or an interval of R) to S. If r is a fixed element of S this map is defined by a unique map t → A(t) from R (or an interval of R) to D such that s(t) = A(t).r. According to Definition 4.1 the motion is C 1 (resp. C 2 ) if and only if the map t → A(t) is C 1 (resp. C 2 ). Then, according to Theorem 1.13, at every time t, there are V ∈ D and W ∈ D such that d d A = V ◦ A or A = A ◦ W with V = A∗ W dt dt V is independent of the choice of r but W depends on this choice. V and W will be called Eulerian and Lagrangian velocity fields of the body. When the motion is C 2 , d ˙ = d W of the maps t → V (t) or W (t) from R (or the derivative V˙ = V and W dt dt an interval of R) to D are well-defined. E XERCISE 4.1.– For fixed r ∈ S the map ν r : D → S by ν r (g) = g.r is a bijection −1 (remind why). Prove that if r and r are fixed in S the maps ν −1 r  ◦ ν r and ν r ◦ ν r  are two reciprocal bijections from D onto D and that they are differentiable. H INT.– Show that those maps are right translations in D and use the results of chapter 1. E XERCISE 4.2.– We refer to definition 4.1. 1) Prove that if, for one fixed r ∈ S, the map Φr is C 1 then, for all r ∈ S, this map is C 1 . 2) Prove that if, for one fixed r ∈ S, the map Fr is C 1 then, for all r ∈ S, this map is C 1 . E XERCISE 4.3.– * This exercise requires some knowledge about topology and manifolds. 1) Check that the following properties are equivalent (notation of exercise 4.1): a) There exists one r ∈ S such that ν r is an diffeomorphism. b) For all r ∈ S the map ν r is a diffeomorphism. 2) Prove that if D acts freely and transitively on S, then there exists a unique topology and a unique manifold structure on S such that properties (a) and (b) are verified.

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3) Check that if (x1 , . . . , x6 ) are coordinates defined on an open subset U of D then (ξ1 , . . . , ξ6 ) defined by ξk (s) = xk (g) if s = g.r,

k = 1, . . . , 6,

are coordinates on the open set ν r (U ) of S. Prove that all coordinate systems of the manifold S for the structure defined in 2) may be constructed by this way. 4.4. Relations to the models of bodies As it was mentioned, a straightforward presentation of the laws of rigid body dynamics on the base of properties [4.1] is available in [CHE 04]. Although we wish to express kinematics and dynamics in a mathematical frame refering only to the Lie group structure of D (and the related manifold structure of S) we think it is worth explaining the links of this mathematical expression with common models of bodies. 4.4.1. Example 1 We assume that D = D(E) is the displacement group of the common three dimension Euclidean affine space E. A model of solid is based on the following assumptions: (S1 ) To every point s ∈ S is associated a subset {s} ⊂ E such that for all g ∈ D and s ∈ S: {g.s} = g {s} . (The set {s} is the concrete representation in E of the configuration s. If we define the action of D on the set P(E) of the subsets of E by the natural way this, property means that the map s → {s} is equivariant.) (S2 ) A particle of the body is defined by a map P : S → E such that for all g ∈ D and s ∈ S:   Ps ∈ {s} and Pg.s = g Ps (In a sense, P is the “name” of a particle regardless of its position in space, Ps is the position of the particle P when the body is in configuration s. If we consider that s ∈ S and {s} are one and the same thing, the action of D will be free, if the subset {s} contains a least three points not on a straight line!). More generally we say that “a point M ” is linked withthe body if, indeed, we have  an equivariant map s → Ms from S to E (i.e. MA.s = A Ms for s ∈ S and A ∈ D). An example will be the center of inertia which is a point linked with a body whose positions are not necessarily those of a physical particle.

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4.4.2. Example 2 A concrete example of principal homogeneous space, often used in mechanics of rigid body, is the space of right-handed orthonormal affine frames of the oriented Euclidean affine space E. Then the space S is: S = {s = (a; e1 , e2 , e3 ) | a ∈ E, (e1 , e2 , e3 ) right-handed orthonormal basis of E} and the action of D on S is the following:   D.s = D(a); D(e1 ), D(e2 ), D(e3 ) with D = linear part of D   Since D ∈ SO(E) when D ∈ D, D(e1 ), D(e2 ), D(e3 ) is a right-handed orthonormal basis of E if (e1 , e2 , e3 ) is. Therefore an action of D on S is well defined and it is easy to prove properties 4.1. Then, given a rigid body, we consider a frame linked with the body, so that it is equivalent to describe the motions of the body and of the frame. Then a physical particle of the body (or a point linked with the body) is defined with respect to the origin of the frame by three coordinates ξ1 , ξ2 , ξ3 , independent of s, such that −−→ aPs = ξ1 e1 + ξ2 e2 + ξ3 e3 when s = (a; e1 , e2 , e3 ) The position of the particle in the configuration D.s is given with respect to the new origin  −−→ −−−−−−→ D(a)PD.s = ξ1 D(e1 ) + ξ2 D(e2 ) + ξ3 D(e3 ) = D aPs 4.4.3. Fundamental theorem of kinematics of a rigid body We now study a motion of the rigid body, observed with respect to some frame of reference and described by a map t → s(t) from I (an interval of R depicting the time) to S. Then the motion of every particle P of a rigid body in the Euclidean space E is given by the map t → Ps(t) . Its velocity and acceleration are expressed according to the following Theorem relating the model of example 1 and the wellknown presentation of kinematics to the assumptions [4.1]: T HEOREM 4.1.– Let a motion of a rigid body observed with respect to some frame of reference, i) If the map t → s(t) is differentiable, there exists a map t → V (t) from R to D such that at every time t ∈ I the velocity of the particle P belonging to the body is expressed by:   vP (t) = V (t) Ps(t) .

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ii) If this map is twice differentiable and t →  V˙ (t) is the derivative of the map I → D defined in i) then the acceleration of P is expressed by:     γ P (t) = V˙ (t) Ps(t) + ω V (t) ∧ V Ps(t) . In kinematics, the dependence of V on time is generally not mentionned and those formulas read vP = V (P ),

γ P = V˙ (P ) + ω V ∧ V (P )

Note that the field of accelerations of the particles of a rigid body is neither reduced to the skew-symmetric field V˙ = V˙ (t) nor a skew-symmetric field. However we shall prove in Chapter 7 that only operations on V and V˙ get involved in dynamics4. 2 The velocity and acceleration of the particle at a fixed time to ∈ I according to classical kinematics are:     d d vP (to ) = , γ P (to ) = Ps(t) v (t) dt dt P t=t0 t=t0 There exists a uniquely defined map τ → A(τ ) from R to D such that s(to + τ ) = A(τ ).s(to ) then, according to [4.1] and (S2 ), A(0) = e, Ps(to +τ ) = A(τ ).Ps(to ) and, according to Definition 4.1, the differentiability of t → s(t) at to is equivalent to the differentiability of τ → A(τ ) at τ = 0: 

d vP (to ) = A(τ ).Ps(to ) dτ  Therefore if V (to ) =

 τ =0

d A(τ ) dτ

Chapter 1, Proposition 1.11): 

d A(τ ).p V (to )(p) = dτ

 τ =0

∈ D is the vector field defined by (see

 τ =0

for all p ∈ E

  we have vP (to ) = V (to ) Ps(to ) . If the motion is differentiable at every time we can  define a map t → V (t) from I to D such that vP (t) = V (t) Ps(t) for every t. When 4 There is a “velocity-screw” but there is no “acceleration-screw”. However dynamics of a rigid body may be expressed with the two elements V and V˙ of the Lie algebra D, corresponding to screws.

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it is twice differentiable we can define the derivative of this map (as a map from I to the vector space D):   d V˙ (to ) = V (t) dt t=to To calculate the acceleration of the particle we use i) at to + τ and write     vP (to + τ ) = V (to + τ ) Ps(to +τ ) = V (to + τ ) A(τ )Ps(to ) ∼

then, since the linear part of V (to ) is ω V (to ):       d = V˙ (to ) Ps(to ) + ω V (to ) ∧ V (to ) Ps(to ) . vP (to + τ ) dτ τ =0 4.5. Changes of frame in kinematics A kinematic frame of reference F is a “laboratory” with an “observer” using some equipments to measure durations and to locate the objects in space. Every “observer” translates what he observes in an abstract language. This language is common to all the observers and they use it to communicate the results of their experiments one to the other. In Newtonian mechanics the observations of the events regarding a physical particle are described by points (t, x) of the space R × E (spacetime) where E is the Euclidean space; for every observation, t is the time, x is the position of the particle relative to the frame. If F1 and F2 are frames of reference there is a derivable (say C 2 ) map t → A12 (t) from R to D such that if a particle is observed as (t1 , x1 ) and (t2 , x2 ) ∈ R × E in F 1 and F 2 respectively, then: t2 = t1 ,

x2 = A12 (t).x1 .

The observations of the events regarding a physical rigid body are described by points of the space R × S where S is defined in section 4.3. If this body is seen as (t1 , s1 ) and (t2 , s2 ) in F1 and F2 respectively, then t2 = t1 ,

s2 = A12 (t).s1 .

Equality t1 = t2 means that all the observers may use similar clocks to measure time from the same origin, property A12 (t) ∈ D for all t means that they use similar devices to measure of lengths and angles in “space” (we neither consider the changes of origin or of unit of time nor the changes of measure of length what would be unimportant in the following). Kinematics needs no more assumptions than differentiability of the map t → A12 (t). Newton’s dynamics relies on more accurate assumptions, in particular the existence of an absolute time an absolute space and of a particular class of frames of references (see Chapter 7).

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T HEOREM 4.2.– Let F1 and F2 be kinematic frames and U12 = U12 (t) be deduced from Theorem 1.13 from Chapter 1 with At = A12 (t) by   d A12 (x) = A12 U12 (x) for x ∈ E. dt Let t → s1 (t) and t → s2 (t) describe the motion of a rigid body with respect to these frames, t → V1 (t) and t → V2 (t) be the velocity fields defined in Proposition 4.1, then at every time s2 = A.s1 ,

V2 = A∗ .(V1 + U12 ),

V˙ 2 = A∗ .(V˙ 1 + U˙ 12 + [U12 , V1 ])

[4.2]

In [4.2] V1 is, at the time t, the field of velocities of the particles of the body with respect to the frame F1 , A∗ U12 (x) is the velocity with respect to F2 of a particle fixed with respect to F1 and in position x ∈ E, in other words the vectorfield A∗ U12 ∈ D describes the induced velocity of F1 with respect to F2 (in French: “vitesse d’entrainement de F1 relativement à F2 ”). 2 By definition s2 = A12 .s1 ≡ A.s1 (from now on, the dependence on time will not be mentionned).  A particle P of the body is seen as Ps1 ∈ E with respect to F1 and as Ps2 = A Ps1 ∈ E with respect to F2 . Therefore on the one hand, by definition of V2 :   d P s = V2 Ps 2 dt 2 and, on the other hand, with Theorem 1.13, Chapter 1

 

d d d   P s2 = A P s1 = A P s1 + A U P s1 dt dt dt  

 

= A V1 Ps1 + A U Ps1 

 = A V1 + U12 (Ps1 )  

 = A V1 + U12 A−1 (Ps2 ) = A∗ (V1 + U12 )(Ps2 )

Since this relation is verified for all particle (and it is assumed that the body is not concentrated on a line...), the second formula [4.2] follows. The last formula is a consequence of the second part of Theorem 1.13, Chapter 1: d V˙ 2 = A∗ (V1 + U12 ) dt   = A∗ (V˙ 1 + U˙ 12 ) + A∗ [U12 , V1 + U12 ] = A∗ (V˙ 1 + U˙ 12 + [U12 , V1 ]).

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R EMARK 4.1.– The algebraic properties of the relations between velocities and accelerations in various frames are the object of exercises 4.4 and 4.5. They were also detailed in reference [CHE 04] (where what is now denoted by U12 (t) was denoted by W12 (t)). For three kinematic frames F1 , F2 and F3 , we have the relation A13 (t) = A23 (t) ◦ A12 (t) and for two frames A12 (t) ◦ A21 (t) = e (identity of D). E XERCISE 4.4.– Prove that, for three kinematicframes F1 , F2 , F3 the induced velocities verify the following relations (where all the quantities are time dependent): U13 = U12 + A21 ∗ U23 ,

U˙ 13 = U˙ 12 + A21 ∗ U˙ 23 + [A21 ∗ U23 , U12 ].

U21 = −A12 ∗ U12 .

[4.3] [4.4]

E XERCISE 4.5.– In this exercise, only kinematics is concerned. Complements fitting with dynamics will be introduced in section 7.1. Let us define the operation in D×D and the left-action of D on D × D by: (X, X  ) (Y, Y  ) = (X + X  , Y + Y  + [X, Y ]),   A∗ .(X, X  ) = A∗ .X, A∗ .X  1) Check that, endowed with operation , D × D becomes a non-commutative group denoted by D(2) and that, in this group, the identity element is (0, 0) and the inverse is expressed by (X, X  )−1 = (−X, −X  ). 2) Check that, for fixed A ∈ D, A∗ : (X, X  ) → A∗ .(X, X  ) is an endomorphism of the group D(2) . 3) Check that relations [4.2] and [4.3] also read

  V2 , V˙ 2 = A12 ∗ . (U12 , U˙ 12 ) (V1 , V˙ 1 ,

[4.5]

(U13 , U˙ 13 ) = A21 ∗ .(U23 , U˙ 23 ) (U12 , U˙ 12 ).

[4.6]

4.6. Graphs and systems subjected to constraints 4.6.1. A few elements of graph theory A (directed) graph on a set B is the structure defined by a subset U ⊂ B × B. Then the members of B are called the vertex of the graph and they will be denoted by Latin letters. The pairs (a, b) ∈ B × B, in this order, such that (a, b) ∈ U are called the arcs of the graph and they will be denoted by Greek letters. The vertice a and b are called origin and end of the arc α = (a, b). D EFINITION 4.2.– A graph is said to be symmetric if (a, b) ∈ U ⇐⇒ (b, a) ∈ U.

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D EFINITION 4.3.– A path in a graph (B, U) is – either an ordered sequence of arcs γ = (α1 , α2 , . . . , αν ) with ν ∈ N∗ , αk ∈ U for k = 1, . . . , ν and if ν > 1, for all k = 1, . . . , ν − 1: origin of αk+1 = end of αk , if ν = 1, γ = (α1 ) is merely the arc α1 of the graph, or a loop of the form γ = (a, a) ∈ U, The origin of γ is the origin of α1 the end (or extremity) of γ is the end of αν . – A cycle (or circuit) of the graph is either a path γ = (α1 , α2 , . . . , αν ) such that origin of α1 = end of αν or a loop of the form (a, a) ∈ U. The set of the paths of the graph (B, U) will be denoted by Path (B, U). – A graph is said to be (strongly) connected5 if for all pair of vertice (a, b) with a = b there is a path γ = (α1 , α2 , . . . , αν ) such that origin of α1 = a,

end of αν = b.

4.6.1.1. Opposite of arcs or paths When a graph is symmetric a natural map U → U : α → α such that (a, b) = (b, a) is defined on the arcs. This map extends into a map Path (B, U) → Path (B, U) such that      γ = αν , . . . , α1 if γ = α1 , . . . , αν , γ = (a, a) if γ = (a, a). Then α and γ are the opposite of the arc α and of the path γ.

5 Since we only consider symmetric graphes there is no practical difference between strong connectivity – defined for directed graphs – and connectivity – defined for non directed graphs –. So we shall speak of connected graph....

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4.6.1.2. Composition of the paths If γ and δ are paths such that the end of γ equal the origin of δ we can define γ ∗ δ as: ⎧ γ ∗ δ = (α1 , . . . , αν , β1 , . . . , βμ ) if γ = (α1 , . . . , αν ), δ = (β1 , . . . , βμ ), ⎪ ⎪ ⎨ (a, a) ∗ δ = δ if δ = (β1 , . . . , βμ ), a = origin of β1 , γ ∗ (a, a) = γ if γ = (α1 , . . . , αν ), a = end of αν , ⎪ ⎪ ⎩ (a, a) ∗ (a, a) = (a, a). In other words, the path δ is put at the end of the path γ. Operation ∗ verifies: γ1 ∗ (γ2 ∗ γ3 ) = (γ1 ∗ γ2 ) ∗ γ3 , (operation ∗ is associative), (γ ∗ δ) = δ ∗ γ. Note that if γ1 ∗ (γ2 ∗ γ3 ) is defined, then (γ1 ∗ γ2 ) ∗ γ3 is also defined. Of course, operation ∗ is not commutative (if γ ∗ δ is defined δ ∗ γ may not be defined). The reason why symmetric directed graphs are interesting is the following: When (B, U) is a symmetric graph and G is a group we may define a symmetric map  −1 X : U → G by the condition that Xα¯ = Xα . When G is a commutative group, with additive notation, we read ∀α ∈ U : Xα = −Xα . (Symmetric maps are involved in some physical laws as “action and reaction” law in dynamics.) Moreover, every symmetric map from X : U → G has a natural extension into a symmetric map also denoted by X : Path (B, U) → G and defined as 

Xγ = Xα1 .Xα2 . . . Xαn if γ = (α1 , α2 , . . . , αn ) Xα = e if γ is a loop (a, a)

then it is readily proved that: Xγ = Xγ−1 for γ ∈ Path (B, U), Xγ1 ∗γ2 = Xγ1 .Xγ2 for γ1 , γ2 ∈ Path (B, U).

4.6.2. The position space of a rigid body system First of all, a system of rigid bodies is described by the list B of the bodies making up the system, so that B is a finite set and a member a of B is the “name” of a body belonging to the system. In practical situations B could be taken as a sequence {1, 2, . . . , n}. When a frame of reference has been fixed, disregarding the possible

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constraints, the positions of a system will be described in a space S = a∈B Sa where, for every a, Sa is the space where the positions of body a described as in section 4.3: a position (also called a configuration) of the system will be defined by a family s = (sa | a ∈ B) (or a sequence (s1 , s2 , . . . , sn )). Since, for each a ∈ B, there is a left action of the group D on Sa verifying properties [4.1] there is also a natural action of D on S = a∈B Sa defined by:     D. sa a∈B = D.sa a∈B However, this action is not transitive in general. In concrete terms the action of D on S describes the displacement by D of the whole system as a rigid system (displacements of the “frozen system”). To describe both a system and the constraints on the relative positions of the bodies of the system we consider a graph (B, U) where – The set B of the vertice of the graph is the list of the bodies making up the system, – The set U (⊂ B × B) of the arcs describes the linked pairs of bodies (the kind of each link - the mechanical device making the joint - has to be otherwise specified), We always assume that: – The graph is symmetric and with no loop: α ∈ U =⇒ α ∈ U and ∀a ∈ B : (a, a) ∈ / U. – The graph is connected. At this stage cycles in the graph might exist. Reasons to consider a symmetric (directed) graph are the following: whereas the mathematical picture often refers to pairs in some order existence of a link between two bodies is obviously a property of a pair of bodies in any order. Moreover, in dynamics, a link between bodies a and b brings about an interaction between them and it is described (in Eulerian picture) by two forces (torsors) Fα ∈ D∗ and Fα ∈ D∗ with α = (a, b); Fα is the force exerted on the body a origin of the arc and Fα is the force exerted on the body b origin of α (i.e. the end of α). When this will be more convenient we shall note Fab = Fα , Fba = Fα . The principle of action and reaction (third Newton’s law) means that Fα + Fα = 0,

(or Fab + Fba = 0)

Note that, this expression of the principle of action and reaction is specific to Eulerian picture.

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4.6.3. The various kinds of links between pairs of bodies Kinematics and dynamics of systems subject to holonomic constraints, in particular the form of definition 4.4 below, were presented in references [CHE 86], [CHE 92]. The positions of a pair (a, b) of bodies may be described in the space Sa × Sb and there exist two left-actions on this space. The left-action of D such that D.(sa , sb ) = (D.sa , D.sb ) (a non-transitive action describing the displacements of the two bodies as if they would make a single rigid body) and the transitive left-action of the group D × D on Sa × Sb such that (A, B).(sa , sb ) = (A.sa , B.sb ) describing all the displacements of the two bodies as if they were independent. 4.6.3.1. Links (Holonomic constraints). Kinematic pairs Here is the definition of a time-independent constraint we shall refer to in the following: D EFINITION 4.4 (Links).– A link (or holonomic constraint) between bodies a and b (in this order) is a relation L in the space Sa × Sb such that sa Lsb and D ∈ D =⇒ D.sa LD.sb A configuration (sa , sb ) of the pair is said to be kinematically admissible (for the link) when sa Lsb is verified. In the following we always assume that there exists almost one kinematically admissible configuration for any linked pair of bodies we consider. The symbol sa Lsb may take two values “true” (when the conditions for the constraint are verified by the configuration (sa , sb )) or “untrue” (when the conditions are not); in all the reasonings, we shall understand “sa Lsb ” as “sa Lsb is true”. Definition 4.4 means that when the pair of bodies is moved as a single rigid body – i.e. the two bodies are subjected to the same displacement – then the constraint is preserved. We keep the word “joint” for a device while “link”, as defined here, has a mathematical meaning. R EMARK 4.2.– Definition 4.4 may be surprising. In standard textbook on kinematics a time independent holonomic constraint would be defined as a relation of the form f (sa , sb ) = 0 where f is some regular function from Sa ×Sb to R (or to Rk ). However when the two bodies are moved by the same displacement the constraint must be preserved what is meaning that f (D.sa , D.sb ) = f (sa , sb ) (a property verified in all concrete examples). Then the following definition of L agrees with definition 4.4: sa Lsb ⇐⇒ f (sa , sb ) = 0 In the frame of definition 4.4,when sa Lsb is verified, let us define the set L(sa , sb ) = {D ∈ D | sa L(D.sb ) (is true).}

[4.7]

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When it is defined, L(sa , sb ) is the set of the displacements of body b allowed by the link when body a remains fixed and we have e ∈ L(sa , sb ). The following property is an evident consequence from definition 4.4: Let κ = (ra , rb ) be a configuration (“reference” configuration) of the pair of bodies a and b such that ra Lrb and let (sa , sb ) = (A.ra , B.rb ), with A and B in D, be any configuration of the pair of bodies. Then a necessary and sufficient condition for sa Lsb is that A−1 .B ∈ L(ra , rb ). E XERCISE 4.6.– 1) Prove that the following properties are consequences from definition 4.4: If D ∈ D and L(sa , sb ) then L(D.sa , D.sb ) = IntD.L(sa , sb ), = L(sa , sb ).D

−1

,

L(sa , D.sb ) L(D.sa , sb ) = D.L(sa , sb )

[4.8]

2) Prove that the following properties are equivalent a) For one configuration (sa , sb ), with sa Lsb : L(sa , sb ) is a Lie subgroup of D b) For all the configurations (sa , sb ), with sa , Lsb : L(sa , sb ) is a Lie subgroup of D And that, when they are verified and sa Lsb , A.sa LB.sb (with A, B ∈ D) then: L(A.sa , B.sb ) = A.L(sa , sb ).A−1 = B −1 .L(sa , sb ).B 3) Assuming that property (a) of the previous question is verified we note l(sa , sb ) the Lie subalgebra of L(sa , sb ). Prove that sa Lsb , A.sa LB.sb , l(A.s, B.sb ) = A∗ l(sa , sb ) = B∗−1 l(sa , sb ). 4) In a sense, when a pair of bodies is considered, the order of the bodies is arbitrary and it may be reversed. As in definition 4.4 define L on Sb × Sa by sb Lsa ⇐⇒ sa Lsb .

[4.9]

Prove that, in general, the space L(sb , sa ) = {D ∈ D | L(sb , D.sa )} associated with L as in [4.7] verifies L(sb , sa ) = L(sa , sb )−1 . and, under the assumptions (a), L(sb , sa ) = L(sa , sb ). H INT.– For 2) remark that, L(sa , sb ).(A−1 B) = L(sa , sb ).

when

A.sa LB.sb ,

(A−1 B).L(sa , sb )

=

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145

There are many kinds of sets L(sa , sb ). If L(sa , sb ) = {e} the two bodies make a single rigid body, if L(sa , sb ) = D there is no effective constraint between a and b. In practice L(sa , sb ) will be at least a submanifold of D. Here we shall assume (see exercise 4.6): D EFINITION 4.5 (Kinematic pairs).– A kinematic pair is a pair (a, b) of bodies subjected to a link such that, for all kinematically admissible configuration (sa , sb ), the set L(sa , sb ) is a Lie subgroup of D. For a kinematic pair L(sb , sa ) = L(sa , sb ). Kinematic pairs have very particular properties which will be explained in Proposition 4.1 below. D EFINITION 4.6 (Field of subgroups on Sa ).– Let S be a space with a left-action of D as in [4.1], a (equivariant) field of (Lie) subgroups on the space S is a relation s → L(s) such that i) For all s ∈ S: L(s) is a Lie subgroup of D. ii) For all s ∈ S and all D ∈ D: L(D.s) = IntD.L(s). To a field of Lie subgroups of D on S is associated a field of Lie subalgebras of D: let l(s) be the Lie algebra of the subgroup L(s) (Definition 1.10) then (Corollary 1.3) these Lie subalgebras of D verify the equivariance property l(D.s) = D∗ l(s) for s ∈ S, D ∈ D P ROPOSITION 4.1.– If bodies a and b are linked by a kinematic pair there is a field of subgroups sa → L(sa ) defined on Sa such that ∀sa ∈ Sa : L(sa , sb ) = L(sa ) when sa Lsb . In particular, if κ = (ra , rb ) is a kinematically admissible reference position, all the subgroups L(sa , sb ) = L(sa ) are conjugate subgroups of L(ra , rb ) = L(ra ) and are conjugate one to the other. 2 Let sb , and sb such that sa Lsb , sa Lsb are verified. Then, the uniquely defined Db ∈ D such that sb = Db .sb belongs to L(sa , sb ) and we deduce that g ∈ L(sa , sb ) ⇔ sa L(g.sb ) ⇔ sa L(gDb ).sb ⇔ gDb ∈ L(sa , sb ) ⇔ g ∈ L(sa , sb ) Therefore L(sa , sb ) = L(sa , sb ). Since there exists configurations of the pair such that sa Lsb (see Definition 4.4), we may define L(sa ) for sa ∈ Sa by L(sa ) = L(sa , sb ) for any sb with sa Lsb . Then, with [4.8], L(D.sa ) = L(D.sa , D.sb ) = IntD.L(sa , sb ) = IntD.L(sa ).

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The common joints of mechanical systems are described as in definition 4.5: rotary joints, prismatic joints, helical joints (screws) with the one-parameter groups of D, spherical joint with three parameter groups and so on (see J. M. Hervé [HER 78, HER 82, HER 94, HER 99]). Reuleaux’s lower pairs (which materialize as a surface sliding on a surface see [SEL 98]) agree with definition 4.5. However Reuleaux’s classification contains only six pairs: rotary, prismatic, helical, cylindrical, spherical and planar pairs corresponding to six possibilities of non-trivial Lie subgroups of D among ten (see Chapter 1 Appendix 2, those six kinds of groups are marked with an asterisk). The kinematic pairs corresponding to the one-parameter groups will be called H-pairs (“helical pair”), in particular R-pairs (“rotary pair”) or P-pairs (“prismatic pair”), according to the subgroups L(sa ) defined in Proposition 4.1 are general oneparameter groups of screwings or, in particular, one-parameter groups of rotations or of translations.

4.6.4. Kinematics of a linked pair of bodies This section introduces the kinematical magnitudes involved in the constitutive law of rheonomic links (Theorem 8.1 below). If there is a link acording to definitions 4.4 and 4.5 between bodies a and b, if κ = (ra , rb ) is a kinematically admissible reference position and L(κ) = L(ra , rb ), then all other configuration of the pair of bodies is defined by sa = Da .ra , sb = Db .rb (Da , Db ∈ D) and then:     sa L sb ⇐⇒ Da .ra L Db .rb ⇐⇒ Da−1 .Db ∈ L(κ)

[4.10]

This relation suggests to define, in general, the function Δκ : Sa × Sb → D, depending on the choice of the reference position κ, such that: Δκ (sa , sb ) = Da−1 .Db if sa = Da .ra and sb = Db .rb

[4.11]

This function describes the “relative positions” of body b with respect to body a and it is invariant by the non-transitive left-action of D on Sa × Sb : Δκ (D.sa , D.sb ) = Δκ (sa , sb ) When a link between a and b is given, then: 

 (sa , sb ) is kinematically admissible ⇐⇒ Δκ (sa , sb ) ∈ L(κ)

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147

P ROPOSITION 4.2.– In the framework of [4.11] if the bodies are moving and if (according to Theorem 1.13) Va and Vb are defined at time t by: d Da = Va ◦ Da , dt

d D b = Vb ◦ D b dt   then, if we note Δκ (sa , sb ) the map t → Δκ sa (t), sb (t) , and Δκ (sa , sb ) the linear part of Δκ (sa , sb ): i) There is a mapping t → κ taking its values in D such that, at time t:  d Δκ (sa , sb ) = κ ◦ Δκ (sa , sb ) with κ = Da−1∗ Vb − Va ), dt ii) There is a map t → wκ taking its values in D such that, at time t: d Δκ (sa , sb ) = Δκ (sa , sb ) ◦ wκ , dt

κ = Δκ (sa , sb )∗ wκ

iii) We assume that the bodies are linked by a kinematic pair and that their motion is kinematically admissible, then  and w ∈ l(κ) for all t,  = Δκ ∗ w. Moreover Vb − Va ∈ l(sa ) (the Lie algebra of L(sa )). In other words properties i) and ii) mean that κ ∈ D and wκ are the terms “V ” and “W ” of Theorem 1.13 for the map t → Δκ sa (t), sb (t) . When a reference configuration κ is fixed Δκ (sa , sb ) describes the relative displacement of body b with respect to body a (and the deformation of the link between a and b) when the pair of bodies moves from (ra , rb ) to (sa , sb ), wκ describes the relative velocity of b with respect to a (and the rate of deformation of the link). 2 Part i) is a simple matter of differential calculus easily derived from Corollaries 1.5 and 1.6 (sse also exercise 4.7 below). When the motion is kinematically admissible, let us fix a value to of t. Then t → Δ(t) ◦ Δ(to )−1 = F (t) and t → Δ(to )−1 ◦ Δ(t) = G(t) are (differentiable) curve by e in D, taking their values in the Lie group L(ra ) and such that F (to ) = G(to ) = e, F (t) = Δ(to ) ◦ G(t) ◦ Δ(to )−1 and, at time to : 

d F (t) dt

 t=to

= (to ) ◦ Δ(to ) ◦ Δ(to )−1 = (to )

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d G(t) dt

d (to ) = F (t) dt

 

t=to

t=to

= Δ(to )−1 ◦ Δ(to ) ◦ w(to ) = w(to )  = Δ(to )∗

d G(t) dt

 t=to

= Δ(to )∗ w(to )

Therefore,  and w ∈ l(ra ) (Theorem 1.17, Appendix 2 to Chapter 1). All the preceding results   would be verified if the reference configuration would be κ = sa (to ), sb (to ) . In that case Da = e at time to and we would find find Vb −Va ∈ l(sa ) with i). R EMARK 4.3.– If the reference position κ is changed into χ = D.κ = (D.ra , D.rb ) then Δχ = IntD.Δκ so that χ = D∗ .κ . Moreover when κ is kinematically admissible, for all D ∈ D, χ = (D.ra , D.rb ) too (see Definition 4.4). E XERCISE 4.7.– *Using the general formalism of Lie groups (see Chapter 1, section 1.5) prove that if t → A(t), t → B(t) and Δ : t → A(t)−1 B(t) are differentiable maps from R to D then, if dot means time derivative and ϑr , ϑ are the Maurer-Cartan forms of D then:     ˙ − ϑr (A) ˙ ˙ = Ad A−1 ϑr (B) ϑr Δ     ˙ − ϑr (A) ˙ ˙ = Ad B −1 ϑr (B) ϑ Δ     = Ad Δ.ϑ B˙ − ϑ A˙ Deduce a proof of Proposition 4.2 in that general formalism. E XERCISE 4.8.– Let (a, b) be a kinematic pair of bodies and κ = (ra , rb ) be a kinematically admissible configuration of this pair. We assume that the motion is twice differentiable and refer to the notations of Proposition 4.2 and of exercise 4.5, section 4.5. Check that there exists w and w˙ in l(κ) such that (Vb , V˙ b ) (Va , V˙ a )−1 = Da ∗ Δκ ∗ (w, w) ˙ 4.7. Kinematics of chains We refer to section 4.6.2 and, moreover, we assume that – The graph (B, U) is symmetric with no loop. – The graph (B, U) is connected.

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149

To specify the kinematic of the system we shall assume that: – For each (a, b) = α ∈ U there exist a link Lα between a and b – For (a, b) = α ∈ U and sa Lα sb , Lα (sa , sb ) = L(sa ) is a Lie subgroup of D (so that we only consider links which are kinematic pairs the defined by a fields of Lie subgroups on Sa an in Proposition 4.1). – For all α ∈ U: Lα¯ = Lα . (The third property means that each physical link is independent of the order of the pair of bodies we consider and, with the second property, Lα (sb , sa ) = L(sa , sb )). We consider a kinematically admissible reference position of the system κ = (ra | a ∈ B) (so that, for all (a, b) = α ∈ U, ra Lα rb is verified) and put Lα (κ) = Lα (ra , rb ) (a Lie subgroup of D), lα (κ) = Lie algebra of this group (a Lie subalgebra of D).Then, any other configuration of the system is defined by s = (sa = Da .ra | a ∈ B, Da ∈ D) and the condition for this configuration be admissible reads: Aα = Da−1 .Db ∈ Lα (κ) for all (a, b) = α ∈ U. The mapping α → Aα from U to D will be extended into a mapping γ → Aγ from Path (B, U) to D as in section 4.6.1. Hence it is evident that: Aγ = Da−1 .Db when a = origin of γ, b = end of γ. Aα is the relative displacement of body b with respect to a allowed by a joint associated with a link α = (a, b) and, more genrally, Aγ is the displacement allowed by a chain of joints between a and b. According to the assumption on the graph, for every a and b ∈ B there is at least a path γ = [a, b] = (α1 , . . . , αp ) ∈ Path (B, A) and, in a problem of mechanics, the configurations of the chain of bodies of γ may be completely described by: 1) the displacement Da of the base body a, 2) articular displacements Aα defined for α ∈ U and verifying Aα ∈ Lα (κ) (if necessary we may put Aα¯ = A−1 α to define Aα for all α ∈ U). Then, clearly: Db = Da .Aα1 . . . Aαp = Da .Aγ with Aα ∈ Lα (κ)

[4.12]

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When the system is moving, the displacements Da and Aα are time dependent and, according to Chapter 1, Theorem 1.13 we define the velocities Va , zγ , wα (all ∈ D) by ⎧ d ⎪ ⎪ D = Va ◦ Da for a ∈ B ⎪ ⎪ dt a ⎪ ⎪ ⎪ ⎪ ⎨ d Aα = Aα ◦ wα for α ∈ U ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ d Aγ = zγ ◦ Aγ for γ ∈ Path (B, U) dt

[4.13]

      In other words Va = ϑr D˙ a , zγ = ϑr A˙ γ , wα = ϑ A˙ α (where dot means time derivative). The calculations necessary to kinematics and dynamics require to find the Va (and V˙ a ) as functions of the wα (and w˙ α ) and Vo (and V˙ o ). For γ ∈ Path (B, A), α ∈ γ, we shall put   Aα γ = Aα1 . . . Aαp ∗ when γ = (α1 , . . . , αm ), α = αp and p < m

[4.14]

α so that Aα γ ∈ L(D) (and not in D). Since Aγ = (Aα1 )∗ ◦ · · · ◦ (Aαp )∗ , the so defined operators depend only on the action of D on D (adjoint representation of the Lie group).

The zγ (and z˙γ ) play the role of auxiliary magnitudes and for γ ∈ Path (B, A), α ∈ γ, the relations are the following: zγ =

 α∈γ

z˙γ =

Aα γ .wα



α∈γ

[4.15]

[Aβγ .wβ , Aα .w ] α γ



Aα ˙α + γ .w

[4.16]

β∈γ, β≤α

In formula [4.16] and in the following β ≤ α will mean that the arc β is before or may be equal to β on the path γ, β < α will mean that the arc β is before (and not equal to) α on the path γ (Remind that the next general results are specific of a choice of γ: without more assumptions on the graph (B, U) the path γ betweeen a and b may not be unique). For a, b ∈ B and γ = [a, b] ∈ Path (B, A), we have (for b = a): Vb − Va = Da ∗

 α∈γ

Aα γ .wα

[4.17]

Kinematics of a Rigid Body and Rigid Body Systems

V˙ b − V˙ a = Da ∗

 Aα ˙α + γ .w

α∈γ



[Aβγ wβ , Aα γ .wα ]

151

[4.18]

β∈γ, β≤α

 

 + Va , Da ∗ Aα .w α γ

[4.19]

α∈γ

Remark that, since [Va , Vb − Va ] = [Va , Vb ], [4.19] also reads V˙ b − V˙ a = [Va , Vb ] + Da ∗

 Aα ˙α + γ .w

α∈γ

[Aβγ wβ , Aα γ .wα ]



[4.20]

β∈γ, β≤α

The proofs of these results is the matter of some exercises. E XERCISE 4.9.– Prove formula [4.15, 4.16] and deduce formula [4.17, 4.19, 4.20]. H INT.– The reader may use the elementary form of the differential calculus in the displacement group D of Chapter 1, Theorem 1.13. However a deeper proof may be derived from the general properties of Lie groups presented in Chapter 1, section 1.6 (conclusions) of that chapter. E XERCISE 4.10.– *Deduce a proof of formulas [4.17], [4.19] from the result of exercise 4.8. (This property opens the way to recursive calculations in kinematics and dynamics of tree structured systems) For calculations in dynamics it is convenient to introduce the quantities Wa , for a ∈ B such that ˙ a, V˙ a = Da ∗ W

Va = D a ∗ W a ,

[4.21]

(note that the last relations [4.21] are consequences of the first ones).  From [4.17] and [4.19] we readily deduce that when γ = [a, b], putting Aγ = Aγ ∗ :

 α Wb = A−1 W , + A w a α γ γ α∈γ

˙ b = A−1 W ˙ a + A−1 W γ γ

[4.22]

 α [Wa , Aα ˙α + γ wα ] + Aγ w

α∈γ



[Aβγ wβ , Aα w ] [4.23] α γ

β∈γ, β≤α

α In fact, for α ∈ γ, we may define the operators Aαγ ∈ L(D) by Aαγ = A−1 γ ◦ Aγ so that, when γ = (α1 , . . . , αm ):

   −1  ⎧ ⎨ Aαγ = A−1 αm ∗ ◦ · · · ◦ Aαp+1 ∗ when α = αp with p < m, ⎩

Aαγ = 1

when α = αm .

[4.24]

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Wb = A−1 γ Wa + ˙ b = A−1 ˙ W γ Wa + +





Aαγ wα ,

[4.25]

α∈γ

 [A−1 ˙α γ Wa , Aαγ wα ] + Aαγ w

α∈γ

[Aβγ wβ , Aαγ wα ]

[4.26]

β∈γ, β≤α

R EMARK 4.4.– The operations involved in formulas [4.17] to [4.26] are solely the operations in the Lie algebra D and the adjoint action of D on D. Therefore all the calculations of kinematics of rigid body systems may be expressed with the structure of Δ-module on D exposed in Chapter 2. Operators Da ∗ and Aα γ are Δ-linear, the ˙ ˙ terms Va , Va , Wa , Wa , wα and w˙ α become dual vectors, the Lie bracket becomes the ˙ a , wα extended cross product. When a basis of Δ has been chosen, Va , V˙ a , Wa , W α and w˙ α are expressed by columns of 3 dual numbers ; Da ∗ and Aγ may be expressed 3 × 3 orthogonal matrices of dual numbers. However, when we turn to dynamics, the structure of Δ-module is not so effective because the inertia operators Hs playing a central role in dynamics are not Δ-linear and the calculations cannot be completely performed in the dual number setting.

5 Kinematics of Open Chains, Singularities

5.1. The mathematical picture of an open chain In the previous chapters all the necessary formula for expanding kinematics or dynamics of any tree structured rigid body system were exposed. In the present chapter we consider a rigid body system which is a chain of bodies linking up two bodies a and b. Then, with the notation of section 4.6.1 S = {a = b0 , b1 , . . . , bm = b}, U = {α1 , α2 , . . . , αm , α1 , α2 , . . . , αm } with αi = (bi−1 , bi ), i = 1, . . . m. and we shall put A = {α1 , α2 , . . . , αm }. In section 5.1 it is assumed that a = b (open chain) whereas in section 6 it will be assumed that a = b (closed chain and the graph is a cycle). When a link between two bodies is considered as the occurrence of a constraint restricting their relative motions, this link is an elements of U (or of A). However when the physical device making the link is considered it may also be called a joint. The point of view of kinematics regarding links was exposed in section 4.6.3. In the present case, all the links (joints) associated with the arcs αi will be assumed to be kinematic pairs (Definition 4.5). If r = (r0 , r1 , . . . , rm ) is a kinematically admissible reference configuration of the system, the kinematic of the chain is defined by the data of a family of Lie subgroups Lαi (r) of D. Their Lie algebras will be denoted by lαi (r). All the other configurations of the chain are of the form s = (s0 , s1 , . . . , sm ) with si = Di .ri and such a configuration is kinematically admissible if Da ∈ D,

−1 Di−1 Di = Aαi ∈ Lαi (ri ) for i = 1, . . . , m

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In this chapter we shall study the “shape function” of the system which is the map

f:

m 

Lαi (r) → D : (Aα1 , . . . , Aαm ) ; Aα1 . . . Aαp

[5.1]

i=1

so that Db = Da .f (Aα1 , . . . , Aαm ), with the notation of section 4.6.1, f (Aα1 , . . . , Aαm ) = Aγ , γ = [a, b]. The shape function depends on the choice of a reference configuration, (f = fr even though this is not systematically specified) and an intrinsic geometrical language would refer to the so called structure of principal fiber bundle. The problem which will be considered in the following concerns the calculation of derivatives of the shape function and the calculation of the rank of its first derivative. A related – and more difficult – task is the determination of the range of the shape function: concrete applications of such problems concern the kinematic of manipulators. Whereas the rank issues may be deeply investigated through Lie group language and may lead to interesting and general results, the other problems are often beyond the scope of the Lie group approach. The deep reason is that the nature of the latter problems are global and also lead to sets which are not (in general) differentiable manifold but are real algebraic varieties and semi-varieties. Nevertheless some local questions like the variations of the rank of f or the manifold structure of Im f and its dimension are interesting in themselves. Although some of these issues (even global or non-differentiable questions) will be mentioned about mechanisms in section 6, interesting developments may be included in the current section. Even though it should be possible to expand the calculations and the reasonings with the expression [5.1] of the shape function we take the opportunity of dealing with kinematic chains and of establishing a bridge with a language closer to the common one in kinematics. The issue of the singularities of a kinematic chain is tackled and solved for the so-called weak singular chains where a certain “transversality condition” holds. Stronger singularity types are out of the scope of this book and are often tackled in the frame of algebraic geometry. A last subsection suggests some further developments in the global aspects of singularities of kinematic chains.  E XERCISE 5.1.– Let S = m of the chain endowed i=1 Si be the configuration space  with the left-action of D such that g.s = g.s1 , . . . , g.sm for g ∈ D (describing the motions of the “frozen system”, see section 4.6.2). Check that the shape function is invariant under this action (so that this function depends only on the choice of r).

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155

5.1.1. Articular coordinates When a kinematically admissible reference configuration r is fixed, in order to simplify the notation we shall note Lαi (r) = Li (r), lαi (r) = li (r) and di = dim li (r) for i = 1, . . . , m (according to chapter 1, di = 1, 2, 3, 4 or 6, and di is independent of the choice of r). The chain is then completely defined by r and the family (Li (r))i=1,...,m of Lie subgroups of D, hence, we now take the opportunity of studying open kinematic chains using coordinates and a language closer to the usual one (even though this could not  be necessary  to further investigations). The process consists in choosing a base x1i , . . . , xdi i in each of the subalgebras li (r) and the mapping defined by qi = (qi1 , . . . , qidi ) ; exp(qi1 x1i ) . . . exp(qidi xdi i )

[5.2]

(For the rest of the chapter, exp = expD denotes the exponential map of D). This mapping may be defined from Rdi to Li (r), it induces a coordinate representation of the Lie subgroup Li (r) in the neighborhood of its unit element (in other words the mapping is only a local diffeomorphism from an open neighborhood Ui ⊂ Rdi of 0 in Rdi onto an open neighborhood Vi of the unit element in the manifold Li (r), called second kind canonical coordinates on the group in general Lie groups theory.) Up to some singular configurations, the kinematics of the chain is then completely defined by r and the family (li (r))i=1,...,m of Lie subalgebras of D. Thus, grouping the bases (xki )k=1,...,di for all the links of the chain leads to a family   xγ = (xki )k=1,...,di i=1,...,m = (x1 , . . . , xn ) ∈ hn ,

n=

n 

di

i=1

where h ⊂ D is the Lie algebra generated by the family xγ which is also generated by the family (lαi (r))i=1,...,m . The coordinates qik are then denoted by q = (q1 , . . . , qn ) ∈ Rn . D EFINITION 5.1.– qi = (qi1 , . . . , qidi ) are called the articular coordinates in a neighborhood of r of the joint Li associated to the choice of the basis (xki )k∈Ii and q = (q1 , . . . , qn ) are the articular coordinates of the kinematic chain associated to the choice of the family xγ = (x1 , . . . , xn ) ∈ hn . We point out that the order of the coordinates and the order of the vectors in the family xγ is of high importance. The way to investigate the kinematics of chains is then reduced to the mathematical properties of the family xγ = (x1 , . . . , xn ) ∈ hn which represents γ. Two remarks must be done. First, the significant properties of the kinematic chain must be independent of the previous non-canonical choices of the bases in the various Lie subalgebras li (r) and the associated coordinates in the Lie subgroups Li (r). If he/she feels it is necessary, the reader can systematically check this property; for example,

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the rank of the family xγ = (x1 , . . . , xn ) ∈ hn is independent of these choices. Second, this representation of the kinematics of γ by xγ strongly depends on the reference configuration r of γ. Contrary to the previous one, this dependency may not be removed and it means that this representation is local and must be fundamentally used for local investigations. This is the object of the rest of the section and of the chapter. 5.1.2. The shape function in articular coordinates Using articular coordinates the local expression of the shape function [5.1] of a chain in the neighborhood of a kinematically admissible reference configuration will be defined according to D EFINITION 5.2.– Let r be a kinematically admissible configuration of an open kinematic chain γ = [a = b0 , b = bm ] and (x1 , . . . , xn ) ∈ hn ⊂ Dn the family of vectors associated with γ and r as in section 5.1.1. The (local) shape function of γ in the neighborhood of r is the map f : U ⊂ Rn → D: q = (q1 , . . . , qn ) ; f (q) = exp(q1 x1 ). . . . . exp(qn xn )

[5.3]

where U is a convenient open neighborhood of 0 in Rn . Although the shape function could be obviously defined on the whole space Rn , in kinematics of the associated chain, the following considerations make sense locally and mainly in such a neighborhood U . Although, in practice, the motions in the joints are often constrained by physical joint stops compelling the articular coordinates to remain in bounded intervals, but many calculations remain valid on Rn . This shape function is also called the work function of the chain (see for example [LER 98b]). As it was mentioned above, the work space of the chain may be investigated through this function. For example, if the base a = b0 does not move, using the language of robotics, rm denotes the reference configuration of the end effector when the chain is in configuration r, and the set {f (q).rm | q ∈ A} describes the positions  of the  end effector of the robot when the joints move over some bounded domain A ⊂ Rn taking account of the physical limitations in the joints. The concept of work volume of a robot is also in relationship with the calculation of f (A) and with the description of its structure. An interesting question is then: what is the structure of f (A) in D or of f (A).rm in S. The answer depends on the structure of A itself (open set, closed set, submanifold,. . . ) and on the shape (or work) function itself. Some precisions may be found in [LER 98b]. The tangent map of the (local) shape function is a mapping between tangent spaces f T : T U → T D. Df (q) maps linearly Tq U in Tf (q) D by the map denoted by Df (q)

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157

(induced by f T on each tangent space). The tangent space  T U maybe identified with U × Rn (see section 1.1) and T D with D × D (by v → o(v), ϑr (v) where o(v) is the origin of the tangent vector v and ϑr is the right Maurer-Cartan form, see section 1.5). With these identifications f T is equivalent to the map   (q, u) ; (f (q), ϑr Df (q)(u) For fixed q ∈ U, the map Df (q) is a linear map from {q} × Rn  Rn to Tf (q) D,   n depending on q, and that, for fixed q, ϑr Df (q)(·) is a linear map  from R to D depending on q ∈ U . We may also consider that q ; ϑr Df (q) is a map from U to the vector space L(Rn , D) and then its tangent at q ∈ U will be a mapping from T U  U × Rn to T L(Rn , D)  L(Rn , D) × L(Rn , D). To conclude, the  derivative  of the local shape function is completely determined by the map q ; ϑr Df (q) . P ROPOSITION 5.1.– The (local) shape function is analytic in an open neighborhood U of the origin of Rn . The map f1 : U → L(Rn , D) such that f1 (q)(u) = ϑr (Df (q)(u))) is expressed by f1 (q)(u) =

n 

uk yk (q) for u ∈ Rn

[5.4]

k=1

with, for all k = 1, . . . , n, yk (q) ∈ D is expressed by: yk (q) = Ad (exp(q1 x1 ). . . . . exp(qk−1 xk−1 )xk = exp(q1 x1 )∗ . . . . . exp(qk−1 xk−1 )∗ .xk

[5.5]

Before proving Proposition 5.1, let us remind that a general theorem in differential calculus says that when U is an open subset of Rn , D is a manifold and f : U → D: i) if f has partial derivatives 

d ∂k f (q) = f (q + τ ek ) dτ



 τ =0

∈ Tf (q) D



((ek = (0, . . . , 0, 1, 0 . . . , 0) with 1 at the k-th place), and if these partial derivatives are continuous functions of q on U for k = 1, . . . , n, then f is derivable (or “differentiable”) and its derivative (or “differential”) at q ∈ U is the map Df (q) : Rn → Tq D such that Df (q)(u) =

n  k=1

uk ∂k (q) for u = (u1 , . . . , uk ) ∈ Rn

()

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In other words  Df (q)(u) =

d f (q + τ u) dτ



 τ =0

∈ Tf (q) D



exists for all u ∈ Rn and it is a linear map from Rn to Tf (q) D expressed by (). ii) Conversely, if the map f is derivable at q, the partial derivatives ∂k f (q) exist for k = 1, . . . , n. However, existence of these partial derivatives (and nothing more) does not entail the derivability (differentiability) of the map f . When the manifold D is a Lie group the tangent vectors ∂k f (q) may be described   by the elements (f  (q), Vk (q)) or (f (q), Wk (q)) ∈ D × D, with Vk (q) = ϑr ∂k (q) or Wk (q) = ϑ ∂k (q) according to Theorem 1.13 and continuity of the partial derivative is equivalent to continuity of the map q → Vk (q) (resp. Wk (q)) from U → D. 2 General properties in analysis in Lie groups entails that f is an analytic map: the exponential maps qi → exp qi xi are analytic as well as all the operation laws in Lie groups. In the following we need above all to know that f has derivatives of any order. Let us calculate the partial derivatives ∂k f (q) and the derivative (the mapping will be differentiable because the partial derivative are continuous). 

d ∂k f (q) = f (q + τ ek ) dτ Df (q)(u) =

n 

 τ =0

uk ∂k f (q) for u = (u1 , . . . , un ) ∈ Rn

k=1

Since exp((qk + τ ek )xk ) = exp qk xk ◦ exp τ xk we may read ⎧ f (q + τ ek ) = Ak ◦ exp τ xk ◦ Bk = LAk ◦ RBk ◦ exp τ xk , ⎪ ⎪ ⎪ ⎪ ⎨ A1 = e Ak = exp q1 x1 · · · exp qk xk for 1 < k ≤ n, ⎪ ⎪ ⎪ ⎪ ⎩ Bn = e Bk = exp qk+1 xk+1 · · · exp qn xn for 1 ≤ k < n,

d  f (q + τ ek ) dτ τ =0 

d = LAk ◦ RBk ◦ exp τ xk dτ τ =0

∂k f (q) =

Kinematics of Open Chains, Singularities

T = LTAk ◦ RB k

159

 d   exp τ xk dτ τ =0

T .xk = LTAk ◦ RB k

  The properties of the right Maurer-Cartan form ϑr LTg v = Ad g.ϑr (v),   T   ϑr Rg v = ϑr (v) for g ∈ D, v ∈ T D leads to ϑr ∂k f (q) = Ad Ak .xk , therefore ϑr (Df (q)(u)) =

n 

uk Ad (exp q1 x1 · · · exp qk xk ).xk

k=1

=

n 

uk Ad (exp q1 x1 · · · exp qk−1 xk−1 ).xk

k=1

E XERCISE 5.2.– Let h be the Lie subalgebra of D generated by the family (x1 , . . . , xn ). 1) Prove that for all j and k, all ξ ∈ R exp(ξxj )∗ xk ∈ h 2) Prove that definition [5.5], entails that for all q ∈ U , the Lie algebra generated by the family (y1 (q), . . . , yn (q)) is equal to h H INT  .– See formula [1.106]. Remind that the Lie subalgebra generated by the family xi is the smallest vector subspace of D containing all thr xk and their Lie brackets of any number of elements xi and also that a vector subspace of a finite dimension vector space is a closed subset. 5.1.3. Further developments about the shape function In this section we check the second and third  derivatives of the shape function. If T D is identified with D × D by the map v → o(v), ϑr (v) , the derivative of f at q was expressed in Proposition 5.1 and it is equivalent to calculate the first and second derivatives of the analytic mapping f1 : U → L(Rn , D) such that f1 (q)(u) = ϑr (Df (q)(u)) =

n 

uk yk (q)

k=1

Then, for fixed q, Df1 (q) is a linear map from Rn to L(Rn , D) and the map f2 from U to L2 (Rn , D) is such that, for fixed q, f2 (q) is the bilinear map f2 (q)(u, v) = Df1 (q)(δq)(v) for δq = u and v in Rn .

[5.6]

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More generally we define step-by-step the analytic maps fp : U → Lp (Rn , D) for all p ≥ 1 by: fp (q)(u1 , . . . , up ) = Dfp−1 (q)(δq)(u2 , . . . , up ) with δq = u1

[5.7]

(Remark that the vector spaces L(Rn , Lp−1 (Rn , D) and Lp (Rn , D) are isomorphic.) P ROPOSITION 5.2.– f1 is analytic and if q ∈ U the expression of f2 (q) ∈ L2 (Rn , D) is: Df1 (q)(u)(v) = f2 (q)(u, v) =

n  

u vk [y (q), yk (q)]

[5.8]

k=1